ISSUES IN QUANTUM FIELD THEORY

N. N. Bogoliubov and D. V. Shirkov

I. THE SCATTERING MATRIX

CONTENTS

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
§ 1. Some information from the theory of free wave fields . . . . . . . . . . . . . . . . . . . . . . . . 150
§ 2. Operator expressions and singular functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
§ 3. Basic concepts of the theory of interacting fields . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
§ 4. The interaction Lagrangian and the \(S\)-matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
§ 5. Expansion of chronological products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
§ 6. Feynman rules for calculating matrix elements and transition probabilities . . . . . . . . . 204
Cited literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

INTRODUCTION

The development of quantum field theory in its modern localizable form encounters, as is well known, great difficulties, especially in attempts to go beyond electrodynamics.

In this connection, a critical exposition of the basic propositions of field theory, with a more thorough analysis of the nature of the difficulties arising in it, may be of interest. Such a consideration constitutes the aim of the proposed series of articles, which opens with the present work.

This first article begins with a brief presentation of the scheme for constructing the theory of quantized free fields (§ 1). Next, questions are considered concerning the definition of singular integrable operator functions, which play the principal role in the study of the scattering matrix in the theory of interacting fields (§ 2). In § 4 a method is set forth for introducing the scattering matrix, not based on the Hamiltonian formalism. Its role is played by explicitly formulated physical conditions, for example the conditions of causality, covariance, etc. The article concludes with

we shall consider the technique of expanding chronological products on the basis of Wick’s theorem (§ 5) and its application to the formulation of Feynman rules for calculating matrix elements (§ 6).

In the following articles it is intended to investigate questions of eliminating the divergences of the asymptotic series for the scattering matrix, questions of generalizing the Schrödinger equation, and to set forth the foundations of the theory of Green’s functions.

§ 1. SOME FACTS FROM THE THEORY OF FREE WAVE FIELDS

In the present section we shall briefly consider the scheme for constructing the quantum theory of free fields, focusing attention on certain fundamental points that we shall need in the further exposition of the main material.

As is known, the quantities playing the basic role in field theories are field functions satisfying linear differential equations—the so-called field equations.

These equations, in each concrete case, may be obtained as Euler equations from the stationarity principle for a certain quadratic functional called the action. From the requirement of invariance of the action with respect to four translations, six Lorentz rotations, and a gauge transformation of the first kind (rotation of the phase factor), the corresponding integrals of motion can be obtained: the 4-vector of energy-momentum, the angular-momentum tensor, and the charge density (Noether’s theorem).

The indicated method of constructing the field equation and the basic dynamical variables, based entirely on the action and its 4-density—the Lagrange function, in contrast to the Hamiltonian formalism, which is based on the Hamiltonian function and canonically conjugate quantities, is called the Lagrangian formalism. An important advantage of the Lagrangian formalism over the Hamiltonian one is its manifest relativistic covariance and complete symmetry of the spatial and temporal coordinates.

Let us consider the transition from the classical theory of a field to the quantum theory. The classical description of a field is characterized by the fact that the theory does not explicitly consider the processes of creation and annihilation (more precisely, of emission and absorption) of particles corresponding to these fields. The concept of the classical field function gives the probability of finding a field particle in a given state. In quantum theory, or the theory of second quantization, the field functions describe aggregates of particles, the processes of mutual transformations of which are explicitly contained in the theory. In connection with this, the field functions of quantum

fields acquire an operator meaning and decompose into particle creation and annihilation operators, between which commutation relations are established. The operator wave functions are determined by the field equations and commutation relations up to a unitary transformation.

The field functions, in this way, are no longer functions in the classical sense (c-functions), but become operators ($q$-functions) acting on the wave function, common to all fields, of secondary quantization, which we shall call the state amplitude*). In the second-quantized theory the mean values and probabilities of states are given by quadratic forms of the state amplitude.

Whereas the transformation properties of field functions under Lorentz transformations are determined by their tensor dimensionality, the transformation of the state amplitude $\Phi$ under Lorentz transformations is carried out with the aid of a unitary operator $U_L$

\[ \Phi \to \Phi' = U_L \Phi . \tag{1.1} \]

Considering an infinitesimal transformation of the coordinate system

\[ \begin{gathered} x \to x' = Lx = x + \delta x;\qquad \delta x^k = a^k + \sum_l g^{kl}x^l\omega^{kl},\\ \omega^{kl} + \omega^{lk} = 0,\qquad l,k=0,1,2,3, \end{gathered} \tag{1.2} \]

where $a^k$ and $\omega^{kl}$ are infinitesimal parameters of four translations and six rotations, $x^0$ is the time variable, $x^\alpha$ ($\alpha=1,2,3$) are the spatial coordinates, and $g$ is the Minkowski tensor:

\[ g^{ik}= \begin{cases} 0\ (i\ne k);\\ 1\ (i=k=0)\\ -1\ (i=k=1,2,3). \end{cases} \]

From the correspondence principle we find that

\[ U_L = 1+\delta U_L, \]

\[ i\delta U_L=\sum_k g^{kk}P_0^k a^k+\frac12\sum_{k,l}g^{kk}g^{ll}M_0^{kl}\omega^{kl}, \tag{1.3} \]

where $P_0^k$ and $M_0^{kl}$ are, respectively, the four-vector of energy-momentum and the tensor of angular momentum of the free field. Of course,

*) The state amplitude may be represented as a vector in a certain Hilbert space; for this reason in the American literature it is usually called the state vector.

in quantum consideration the quantities \(P_0\) and \(M_0\) are already operators. In this case they are expressed in terms of the wave operator functions by the same relations that follow from Noether’s theorem as in the classical theory, of course with the establishment of the proper order of the operator factors, which we shall discuss below.

Considering the matrix elements of the field operators with the help of (1.3), we obtain an equation determining the transformation of the wave function \(u(x)\) under a translation of the coordinate systems

\[ ig^{kk}\frac{\partial u(x)}{\partial x^k}=[u(x),P_0^k]_- = u(x)P_0^k-P_0^k u(x), \tag{1.4} \]

where \(P_0^k\) is the operator 4-vector of momentum. Equation (1.4) plays an important role in the process of second quantization. From it there immediately follows the physical meaning of the positive-frequency and negative-frequency parts of operator wave functions\(^1\). Representing the field \(u(x)\) of particles with mass \(m\) in the form

\[ u(x)=u^{(+)}(x)+u^{(-)}(x), \tag{1.5} \]

\[ u^{(+)}(x)=\int_{k^0>0} e^{ikx}\delta(k^2-m^2)u^{(+)}(k)\,dk, \tag{1.6} \]

\[ u^{(-)}(x)=\int_{k^0>0} e^{-ikx}\delta(k^2-m^2)u^{(-)}(k)\,dk, \tag{1.7} \]

where

\[ kx=\sum_i g^{ii}x^ik^i=x^0k^0-\mathbf{kx}; \]

\[ k^2=(k^0)^2-\mathbf{k}^2;\qquad dk=dk^0\,d\mathbf{k}, \]

with the aid of (1.4) we see that if \(\Phi\) is the amplitude of a state with a definite value of the energy-momentum 4-vector \(K^l\),

\[ P^l\Phi=K^l\Phi, \]

then

\[ P^l\bigl(u^{(+)}(k)\Phi\bigr)=(K^l+k^l)\Phi \]

and

\[ P^l\bigl(u^{(-)}(k)\Phi\bigr)=(K^l-k^l)\Phi, \]

where in both cases

\[ k^0=\sqrt{\mathbf{k}^2+m^2}>0. \]

It follows from these equations that the operator \(u^{(+)}(k)\) corresponds to the creation of a particle with mass \(m\) and 4-momentum \(k\), while the operator \(u^{(-)}(k)\) corresponds to its annihilation.

By analogous reasoning, on the basis of the law of transformation of operator functions and state amplitudes under a gradient transformation of the first kind, one can show that in the case of a complex field the operators \(u^{(+)}\) and \(u^{(-)}\) increase the charge of the system by one unit, while the operators \(\dot u^{(+)}\) and \(\dot u^{(-)}\) decrease it by one unit. Therefore the operators of a complex field describe the creation and annihilation of particles of two opposite charges.

Defining the vacuum as a state without particles, we obtain the relation

\[ u^{(-)}(k)\Phi_{\mathrm{vac}}=0, \tag{1.8} \]

which, together with the normalization condition

\[ (\dot\Phi_{\mathrm{vac}},\Phi_{\mathrm{vac}})=1 \tag{1.9} \]

may be regarded as the definition of the vacuum of free fields. It is also obvious that, by acting on \(\Phi_{\mathrm{vac}}\) with various numbers of positive-frequency operators \(u^{(+)}\), we can obtain state amplitudes containing the corresponding number of particles. The general amplitude for an arbitrary state is represented by a superposition of such amplitudes

\[ \Phi=\sum_{(s,j)}\int F_s(k_1,\ldots,k_s)\delta(k_1^2-m_j^2)\ldots \]

\[ \ldots \delta(k_s^2-m_j^2)u_{j_1}^{(+)}(k_1)\ldots u_{j_s}^{(+)}(k_s)\,dk_1\ldots dk_s\,\Phi_{\mathrm{vac}}. \tag{1.10} \]

Passing from (1.10) to the configuration representation, one can obtain the Fock representation for the state amplitude.

From (1.8) there follows the most convenient order of operator factors in expressions for dynamical variables. Requiring that the mean values of these dynamical variables in the vacuum be equal to zero, i.e.

\[ (\dot\Phi_{\mathrm{vac}},P\Phi_{\mathrm{vac}})=0\ \text{etc.}, \]

we obtain that in these expressions all creation operators \(u^{(+)}\) must stand to the left of all annihilation operators \(u^{(-)}\). Such an ordering of operators is called normal. Of course, the product of any number of field operators can be represented as a sum of operator terms in normal form and of some polynomial in the permutation functions. For example,

\[ u(x)u(y)=\bigl(u^{(+)}(x)+u^{(-)}(x)\bigr)\bigl(u^{(+)}(y)+u^{(-)}(y)\bigr)= \]

\[ =u^{(+)}(x)u^{(+)}(y)+u^{(+)}(x)u^{(-)}(y)\pm \]

\[ \pm u^{(+)}(y)u^{(-)}(x)+u^{(-)}(x)u^{(-)}(y)+\frac{1}{i}\Delta^{(-)}(x-y), \tag{1.11} \]

where \(\Delta^{(-)}\) is the result of commutation (or anticommutation—depending on the statistics) of the operators \(u^{(-)}(x)\) and \(u^{(+)}(y)\), i.e.

\[ [u^{(-)}(x),u^{(+)}(y)]_{\mp}=-\frac{1}{i}\Delta^{(-)}(x-y). \]

In complete analogy with how this was done in (1.11), in the more general case the product of a larger number of operators

\[ u(x_1)u(x_2)\ldots u(x_n) \]

can be represented as the sum of products of \(n\) \(\pm\)-frequency operators in normal form and of a certain polynomial \(P\) in the permutation functions \(\Delta^{(-)}\):

\[ \begin{aligned} u(x_1)u(x_2)\ldots u(x_n) &=\sum (-1)^{\eta}u^{(+)}(x_{i_1})u^{(+)}(x_{i_2})\ldots u^{(-)}(x_{i_k})\ldots u^{(-)}(x_{i_n}) \\ &\quad + P\bigl[\Delta^{(-)}(x_i-x_j)\bigr], \end{aligned} \tag{1.12} \]

where the coefficients of the polynomial \(P\), in turn, contain \(m<n\) operators \(u^{(+)}\) and \(u^{(-)}\), and \(\eta\) is the parity of the Fermi permutation in passing from the order \((1,2,\ldots,n)\) to the order \((i_1,i_2,\ldots,i_n)\). Relation (1.12) can be rewritten in the form

\[ u(x_1)u(x_2)\ldots u(x_n)=:u(x_1)u(x_2)\ldots u(x_n):+P\bigl[\Delta^{(-)}(x_i-x_j)\bigr], \]

where the symbol \(:u(x_1)\ldots u(x_n):\) denotes the so-called normal product of operators. In the special case (see (1.11)) we have, by definition,

\[ :u(x)u(y):=u^{(+)}(x)u^{(-)}(y)+u^{(+)}(x)u^{(-)}(y)\pm \]

\[ \pm u^{(+)}(y)u^{(-)}(x)+u^{(-)}(x)u^{(-)}(y). \]

Thus, in the general case, the normal product may be defined either as the zero-order term of the expansion (1.12) in powers of \(\Delta^{(-)}\), or, equivalently, as the ordinary product of the given operators reduced to normal form in such a way that, in the process of reduction, all permutation functions are regarded as equal to zero.

The vacuum mean value of a normal product is always equal to zero. Therefore the rule for obtaining the operator dynamical variables \(P\), \(M\), etc. from the corresponding classical expressions consists in passing in these expressions from ordinary products of classical wave functions to normal products of operator field functions. In this way the meaningless zero vacuum energies, charges, etc., are automatically excluded from the theory.

The permutation relations between operator wave functions are uniquely determined for each field from considerations of the correspondence principle, the requirement of positivity of the observed value of the energy operator, and the structure of equations (1.4). Omitting the corresponding presentation*), we shall describe the structure of the resulting commutation relations, restricting ourselves to the case of complex fields. All positive-frequency operators \(u^{(+)}\) (creation operators) strictly commute (or anticommute) with all creation operators. All annihilation operators \(u^{(-)}\) likewise strictly commute (or anticommute) among themselves, i.e. always

\[ [u^{(+)}(x),u^{(+)}(y)]_{\pm} = [u^{(+)}(x),\dot u^{(+)}(y)]_{\pm} = [\dot u^{(+)}(x),\dot u^{(+)}(y)]_{\pm} =0 \]

and

\[ [u^{(-)}(x),u^{(-)}(y)]_{\pm} = [u^{(-)}(x),\dot u^{(-)}(y)]_{\pm} = [\dot u^{(-)}(x),\dot u^{(-)}(y)]_{\pm} =0. \]

The creation and annihilation operators of particles with different charges also always commute (or anticommute), i.e.

\[ [u^{(+)}(x),u^{(-)}(y)]_{\pm} = [\dot u^{(+)}(x),\dot u^{(-)}(y)]_{\pm} =0. \]

Thus only the following permutations are different from zero:

\[ [u^{(+)}(x),\dot u^{(-)}(y)]_{\pm} = \frac{1}{i}\Delta^{(+)}(x-y), \tag{1.13} \]

\[ [u^{(-)}(x),\dot u^{(+)}(y)]_{\pm} = \frac{1}{i}\Delta^{(-)}(x-y), \tag{1.14} \]

where the functions \(\Delta^{(+)}\) and \(\Delta^{(-)}\), depending on the difference \(x-y\), are respectively the \((+)\)- and \((-)\)-frequency parts of the complete permutation function:

\[ [u(x),\dot u(y)]_{\pm} = \frac{1}{i}\Delta(x-y), \tag{1.15} \]

\[ \Delta(x)=\Delta^{(+)}(x)+\Delta^{(-)}(y). \tag{1.16} \]

In the case where the operators of the given field commute with one another, i.e.

\[ [u(x),\dot u(y)]_{-} \equiv u(x)\dot u(y)-\dot u(y)u(x) = \frac{1}{i}\Delta(x-y), \]

* See, for example, 4.

particles prove to obey Bose–Einstein statistics. In the opposite case

\[ [u(x),\dot u(y)]_{+}=u(x)\dot u(y)+\dot u(y)u(x)=\frac{1}{i}\Delta(x-y) \]

we arrive at Fermi–Dirac statistics and the Pauli principle.

The kind of statistics and the form of the permutation function are determined for each field separately from the considerations already indicated. One obtains:

For the complex scalar field \(\varphi(x)\) (spin 0)

\[ [\varphi(x),\dot{\varphi}(y)]_{-}=\frac{1}{i}D(x-y), \tag{1.17} \]

where \(D(x)\) is the well-known Pauli–Jordan permutation function

\[ D(x)=\frac{1}{(2\pi)^3 i}\int dk\, e^{ikx}\,(k^2-m^2)\,\varepsilon(k^0), \tag{1.18} \]

\[ \varepsilon(k^0)= \begin{cases} +1, & k^0>0,\\ -1, & k^0<0. \end{cases} \tag{1.19} \]

For the complex vector field \(B_n(x)\) (spin 1)

\[ [B_n(x),\dot B_k(y)]_{-} =i\left(g^{kn}+\frac{1}{m^2}\frac{\partial^2}{\partial x^k\partial x^n}\right)D(x-y). \tag{1.20} \]

For the electromagnetic field \(A_n(x)\) (spin 1, mass 0), when quantized with the aid of an indefinite metric \(^{20,21}\),

\[ [A_n(x),A_m(y)]=-ig^{mn}D_0(x-y), \tag{1.21} \]

where

\[ D_0(x)=D(x)\big|_{m=0}=\frac{1}{(2\pi)^3 i}\int dk\,e^{ikx}(k^2)\varepsilon(k^0). \tag{1.22} \]

Finally, for the spinor field \(\psi,\bar\psi\) (spin \(1/2\))

\[ [\psi_\alpha(x),\bar\psi_\beta(y)]_{+}=\frac{1}{i}S_{\alpha\beta}(x-y), \tag{1.23} \]

where

\[ S_{\alpha\beta}(x-y)= \left(i\sum_k\gamma^k\frac{\partial}{\partial x^k}+m\right)_{\alpha\beta}D(x), \tag{1.24} \]

and here we have adopted Feynman’s system of Dirac matrices

\[ \left(i\sum_k\gamma^k\frac{\partial}{\partial x^k}-m\right)\psi(x)=0; \qquad \gamma^m\gamma^n+\gamma^n\gamma^m=2g^{mn}. \tag{1.25} \]

and the adjoint spinor \(\bar\psi\) is defined by the relation

\[ \bar\psi(x)=\psi^\dagger(x)\gamma^0 . \tag{1.26} \]

An important feature of the commutation functions of all fields is their connection with the Pauli–Jordan function \(D(x)\). From (1.17), (1.20), (1.21), and (1.23) it follows that always

\[ \Delta_{\alpha\beta}(x)=P_{\alpha\beta}\left(\frac{\partial}{\partial x}\right)D(x), \tag{1.27} \]

where \(P_{\alpha\beta}\) is a polynomial in the derivatives \(\dfrac{\partial}{\partial x^k}\). In view of the fact that the function \(D(x)\) vanishes outside the light cones, relation (1.27) expresses the independence of events separated by a spacelike interval. Thus all the local wave fields usually considered possess this property.

An important feature of the commutation function \(D(x)\) (and, consequently, in general of all the functions \(\Delta(x)\)) and of its positive- and negative-frequency parts \(D^{(+)}\) and \(D^{(-)}\) is their singular character, expressed in the presence of singularities on the light cone of the type \(\lambda^{-1}\), \(\ln \lambda\), \(\delta(\lambda)\), and finite jumps (here \(\lambda\) is the square of the interval, equal to \(\lambda=x^2=(x^0)^2-(\mathbf{x})^2\)). In view of the fact that matrix elements of various transitions in the theory of interacting fields are expressed through integrals of products of various combinations of the \(\Delta\)-, \(\Delta^{(+)}\)-, and \(\Delta^{(-)}\)-functions, the indicated singularities in individual cases lead to divergent expressions. For a proper understanding of the nature of such infinities, to which, in particular, all ultraviolet catastrophes belong, we shall consider in somewhat greater detail the properties of operator expressions constructed from quantized wave functions of free fields.

§ 2. OPERATOR EXPRESSIONS AND SINGULAR FUNCTIONS

Before proceeding to the exposition of the theory of interacting fields, we shall first have to become acquainted with a number of properties, mainly of an algebraic character, possessed by operator expressions constructed from quantized wave functions of free fields.

Let us take a typical operator expression, depending on the values of the positive- and negative-frequency parts of such functions at a number of space-time points \(x_1,\ldots,x_n\), and represented in normal form

\[ A(x_1,\ldots,x_n)= \sum K_{\ldots\alpha\ldots\beta\ldots}(x_1,\ldots,x_n)\ldots u_\alpha^{(+)}(x_r)\ldots \ldots u_\beta^{(-)}(x_s)\ldots \tag{2.1} \]

In formula (2.1) ... \(u_{\alpha}(x)\) ... are the components of the wave functions, including the conjugate functions and their partial derivatives, and
\(K_{\ldots \alpha \ldots \beta \ldots}(x_{1}, \ldots, x_{n})\) are certain \(c\)-functions of the variables \(x_{1}, \ldots, x_{n}\), which, in view of the homogeneity of space-time, possess the property of translational invariance

\[ K_{\ldots \alpha \ldots \beta \ldots}(x_{1}+a, \ldots, x_{n}+a) = K_{\ldots \alpha \ldots \beta \ldots}(x_{1}, \ldots, x_{n}). \tag{2.2} \]

We shall agree to call these functions the coefficient functions of the given operator expression (2.1). In connection with the singularity of the permutation functions \(\Delta\), they will, generally speaking, be singular, and the question of their mathematical nature will be discussed specially below. For the moment let us note that, by means of the coefficient functions, it is not difficult to express directly the matrix elements of the operator \(A(x_{1}, \ldots, x_{n})\) over all possible states

\[ \Phi_{\ldots \gamma \ldots \rho \ldots} = \ldots u_{\gamma}^{(-)}(p)\ldots \Phi_{\mathrm{vac}}, \tag{2.3} \]

corresponding to the presence of given kinds of particles with prescribed momenta.

For this it is necessary only to determine matrix elements of the type

\[ \left( \Phi_{\ldots \gamma \ldots \rho \ldots} \ldots u_{\alpha}^{(+)}(x_{r}) \ldots u_{\beta}^{(-)}(x_{s}) \ldots \Phi_{\ldots \gamma' \ldots \rho' \ldots} \right) = \]

\[ = \left( \Phi_{\mathrm{vac}} \ldots u_{\gamma}^{(-)}(p) \ldots u_{\alpha}^{(+)}(x_{r}) \ldots u_{\beta}^{(-)}(x_{s}) \ldots u_{\gamma'}^{(+)}(p') \ldots \Phi_{\mathrm{vac}} \right), \]

which is not difficult to carry out with the aid of the permutation relations (1.14), written in the form

\[ \bigl[u_{\alpha}^{(-)}(k),\, u_{\beta}^{(+)}(p)\bigr]_{\pm} = \delta(k-p)P_{\alpha\beta}(k),^{1} \tag{2.4} \]

where \(P_{\alpha\beta}(k)\) is the polynomial corresponding to \(P_{\alpha\beta}\!\left(\dfrac{\partial}{\partial x}\right)\) in (1.26).

Indeed, passing from \(u^{(\pm)}(x)\) to the momentum representation, we shall move the creation operators \(u_{\alpha}^{(+)}(k_{r})\) to the left, and the annihilation operators \(u_{\beta}^{(-)}(k_{s})\) to the right, until they either “annihilate” themselves respectively with \(u_{\gamma}^{(-)}(p)\), \(u_{\gamma'}^{(+)}(p)\), or give zero by acting on the amplitude of the vacuum state. The \(\delta\)-functions that appear in such an annihilation will remove the integrations over \(k_{r}\), making them equal to \(\ldots p \ldots, \ldots, p' \ldots\).

As a result of the indicated operations we arrive at expressions of the form

\[ \bigl(\widehat{\Phi}_{\ldots \tau \ldots \rho \ldots} A(x_1,\ldots,x_n)\Phi_{\ldots \tau' \ldots \rho' \ldots}\bigr) = \]

\[ = \sum_{\ldots \tau \ldots \rho \ldots}^{\ldots \alpha \ldots \beta \ldots} (\ldots \rho \ldots \rho' \ldots) K_{\ldots \alpha \ldots \beta \ldots}(x_1,\ldots,x_n) \times \]

\[ \times \exp\left\{ i\left(\sum_\nu p_\nu x_\nu-\sum_\lambda p_\lambda x_\lambda\right) \right\} I(p_\omega,p'_\omega), \tag{2.5} \]

where the factor

\[ I(p_\omega,p'_\omega)=\prod_\omega \delta(p_\omega-p'_\omega) \]

arises in the case when part of the operators \(u^{(+)}(p'_\omega)\) from
\(\Phi_{\ldots \tau' \ldots \rho' \ldots}\) is paired with the operators
\(u^{(-)}(p_\omega)\) from
\(\widehat{\Phi}_{\ldots \tau \ldots \rho \ldots}\).

In view of the obvious convenience of the normal form of representing operator expressions (2.1), the question of the corresponding method of reduction becomes of interest. It is clear that, in order to reduce to normal form operator expressions depending polynomially on wave functions, it is sufficient to be able to reduce products of the type

\[ A_1(x_1)\ldots A_n(x_n), \]

in which \(A_i(x_i)\) will be linear combinations of the corresponding
\(u_j^{(+)}(x_i)\) and \(u_j^{(-)}(x_i)\). The indicated prescription follows from one important theorem established by Wick \(^{2}\), to the formulation of which we shall now turn.

Let us first consider the case \(n=2\), when we have the product of two linear operators

\[ A_1(x_1)A_2(x_2). \]

According to (1.11), this product can differ from the normal product

\[ :A_1(x_1)A_2(x_2): \]

only by a \(c\)-expression, which we shall call a contraction and denote by means of a bracket below:

\[ A_1(x_1)A_2(x_2) = :A_1(x_1)A_2(x_2): + \underbrace{A_1(x_1)A_2(x_2)}. \tag{2.6} \]

Obviously, one can also define contraction as the vacuum mean of the ordinary product

\[ \contraction{}{A_1}{(x_1)}{A_2} A_1(x_1)A_2(x_2) = \bigl(\Phi_{\mathrm{vac}}^{*} A_1(x_1)A_2(x_2)\Phi_{\mathrm{vac}}\bigr). \tag{2.7} \]

For example, for a spinor field, with the aid of (1.22) we find:

\[ \contraction{}{\psi_\alpha}{(x)}{\psi_\beta} \psi_\alpha(x)\psi_\beta(y) = \frac{1}{i}S_{\alpha\beta}^{(-)}(x-y). \tag{2.8} \]

In an analogous way one can also define contractions of other fields.

In order to formulate Wick’s theorem, it is necessary to introduce the notion of a normal product with contractions. Let us note that from the definition of a normal product (see § 1) follows the commutativity (up to sign) of the operators under the sign of the normal product, i.e.

\[ :A_1(x_1)\ldots A_l(x_l): = (-1)^r :A_{i_1}(x_{i_1})\ldots A_{i_m}(x_{i_m}):, \]

where \(r\) is the parity of the permutation of the operators’ arguments, and also of the normal product with respect to each factor.

We now define the normal product with contractions by taking, by definition, the expression

\[ :\contraction{}{A_1}{(x_1)}{A_2} A_1(x_1)A_2(x_2)A_3(x_3)\ldots \contraction{}{A_k}{(x_k)\ldots A_{n-1}(x_{n-1})}{A_n} A_k(x_k)\ldots A_{n-1}(x_{n-1})A_n(x_n): \]

to be equal to the product of all contractions

\[ \contraction{}{A_1}{(x_1)}{A_3} A_1(x_1)A_3(x_3)\ldots \contraction{}{A_k}{(x_k)}{A_n} A_k(x_k)A_n(x_n) \]

with the normal product of the remaining unpaired operators and with the factor \((-1)^\eta\), i.e.

\[ :\contraction{}{A_1}{(x_1)}{A_2} A_1(x_1)A_2(x_2)A_3(x_3)\ldots \contraction{}{A_k}{(x_k)\ldots A_{n-1}(x_{n-1})}{A_n} A_k(x_k)\ldots A_{n-1}(x_{n-1})A_n(x_n): = \]

\[ = (-1)^\eta \contraction{}{A_1}{(x_1)}{A_3} A_1(x_1)A_3(x_3)\ldots \contraction{}{A_k}{(x_k)}{A_n} A_k(x_k)A_n(x_n) \times \]

\[ \times :A_2(x_2)A_4(x_4)\ldots A_{k-1}(x_{k-1})A_{k+1}(x_{k+1})\ldots A_{n-1}(x_{n-1}):, \tag{2.9} \]

where \(\eta\) is the parity of the permutation

\[ \left( \begin{array}{cccccccccc} 1,&2,&3,&\ldots,&k-1,&k,&k+1,&\ldots,&n-1,&n\\ 1,&3,&\ldots,&k,&n,&2,&4,&\ldots,&k-1,&k+1,\ldots,n-1 \end{array} \right). \]

We can now give a simple formulation of Wick’s theorem. According to this theorem, the ordinary product of linear operators is equal to the sum of all corresponding normal products with all possible contractions, including also the normal product without contractions,

\[ A_1\ldots A_n = :A_1\ldots A_n: + :\underbrace{A_1A_2}\ldots A_n: +\cdots \]
\[ \cdots+ :A_1\ldots \underbrace{A_{n-1}A_n}: + :\underbrace{A_1A_2}\underbrace{A_3A_4}\ldots A_n: +\cdots \tag{2.10} \]

For the proof we shall need the following lemma: if \(A_1,\ldots,A_n,B\) are linear operators, then

\[ :A_1\ldots A_n:B = :A_1\ldots A_nB: + \sum_{1\leq k\leq n} :A_1\ldots \underbrace{A_k\ldots A_nB}: \tag{2.11} \]

Noting that the lemma is obvious for the case when \(B\) consists only of annihilation operators and, consequently,

\[ \underbrace{A_kB}=0, \]

let us consider the case when \(B\) is a creation operator. In this case, obviously, it is sufficient to restrict ourselves to considering \(A_1,\ldots,A_n\) that are annihilation operators. Indeed, if, for example, some \(A_i\) are creation operators, then \(A_iB=0\), and all \(A_i\) can be moved to the left in all normal products, reducing the problem to annihilation operators alone. But in order to represent the expression

\[ :A_1^{(-)}\ldots A_n^{(-)}:B^{(+)} = A_1^{(-)}\ldots A_n^{(-)}B^{(+)} \]

in normal form, it is necessary to commute \(B^{(+)}\) \(n\) times with each of the \(A_i^{(-)}\) \((i=1,\ldots,n)\). As a result we obtain formula (2.11), which is thus proved for the case when the \(A_1,\ldots,A_n,B\) entering into it are each separately either a creation operator or an annihilation operator. But by virtue of the noted property of linearity, lemma (2.11) is thereby also proved for linear operators.

Let us now note that the lemma is directly generalized to the case of normal products with any number of contractions, since the latter, up to \((-1)^q\), is equal to the product of these contractions (\(c\)-functions) by all the remaining uncontracted operators. After this remark, let us proceed to the proof of Wick’s theorem.

It is clear that the theorem is true for \(n=2\). We shall therefore use the method of induction. Assuming that relation (2.10) is true for the product of \(n\) linear operators \(A_i\), we shall try to prove

for the product of \(n+1\) operators. For this, multiply (2.10) on the right by some linear operator \(A_{n+1}\).

But, by the generalized lemma just proved, the normal product of the operators \(A_1,\ldots,A_n\) with any prescribed number of contractions between them, multiplied on the right in the ordinary sense by \(A_{n+1}\), is equal to the sum of the normal products of the operators \(A_1,\ldots,A_{n+1}\), in which, in addition to the contractions already present, all possible (including zero) contractions of free operators from \(A_1,\ldots,A_n\) with \(A_{n+1}\) occur. Thus we see that the ordinary product of \(n+1\) linear operators \(A_1,\ldots,A_{n+1}\) is represented as a sum of normal products with all possible contractions, which completes the induction.

The proof of Wick’s theorem is complete. It is not difficult to see that this theorem is also applicable to the case where some of the factors enter under the signs of normal products

\[ :A_1\ldots A_k::A_{k+1}\ldots A_n:\ldots:A_{s+1}\ldots A_n:. \tag{2.12} \]

In this case Wick’s theorem is formulated in exactly the same way as for a simple product, with the obvious difference that it is now necessary not to count (to set equal to zero) contractions between factors belonging to one and the same normal product.

As an illustration of Wick’s theorem, let us consider the product of two spinor currents. According to (2.10) and taking (2.8) into account, we have:

\[ \begin{aligned} j^k(x)j^l(y) &=:\bar\psi(x)\gamma^k\psi(x)::\bar\psi(y)\gamma^l\psi(y):=\\ &=:\bar\psi(x)\gamma^k\psi(x)\bar\psi(y)\gamma^l\psi(y): +:\bar\psi(x)\gamma^k\psi(x)\bar\psi(y)\gamma^l\psi(y):\\ &\quad+:\bar\psi(x)\gamma^k\psi(x)\bar\psi(y)\gamma^l\psi(y): +:\bar\psi(x)\gamma^k\psi(x)\bar\psi(y)\gamma^l\psi(y):=\\ &=\operatorname{Sp}\{\gamma^k S^{-1}(x-y)\gamma^l S^{-1}(y-x)\}-\\ &\quad-i:\bar\psi(x)\gamma^k S^{-1}(x-y)\gamma^l\psi(y):-\\ &\quad-i:\bar\psi(y)\gamma^l S^{-1}(y-x)\gamma^k\psi(x): +:j^k(x)j^l(y): . \end{aligned} \tag{2.13} \]

Let us now discuss the question of the structure of the coefficient functions which appear when the operation of multiplying operator expressions is performed. Take two arbitrary operators \(A_1(x_1,\ldots,x_n)\) and \(A_2(y_1,\ldots,y_m)\), belonging to the type (2.1) under consideration, with coefficient functions \(K^{(1)}(x_1,\ldots,x_n)\) and \(K^{(2)}(y_1,\ldots,y_m)\), respectively. It is then clear that the product of these operators

\[ A_1(x_1,\ldots,x_n)A_2(y_1,\ldots,y_m) \tag{2.14} \]

also belongs to the type under consideration, and, according to Wick’s theorem, its coefficient functions have the form

\[ K^{(1)}(x_1,\ldots,x_n)\,K^{(2)}(y_1,\ldots,y_n)\prod \Delta^{(-)}(x_j-y_j). \tag{2.15} \]

In view of the fact that the function \(D^{(-)}(x)\), and still more the function

\[ \Delta^{(-)}(x)=P\left(\frac{\partial}{\partial x}\right)D^{(-)}(x) \]

possess a high degree of singularity on the light cone, serious doubts may arise as to whether expressions such as (2.14), (2.15), which include their product in arbitrary number, have any real meaning.

In this connection it is necessary to consider the question of what conditions should be imposed on the coefficient functions of expressions of type (2.1) so that these expressions can be given a definite meaning, or, as one says, so that they contain no “divergences.”

It is clear, first of all, that we cannot require the coefficient functions \(K(x_1,\ldots,x_n)\) to be functions in the sense generally accepted in mathematics, since then we would have to exclude altogether from consideration such “singular” or “improper” functions as \(\delta(x)\), \(D(x)\), etc., with which one constantly has to deal in quantum field theory.

It is therefore natural to regard \(K(x_1,\ldots,x_n)\) as defined precisely as improper functions.

We shall now try to formulate what is normally always meant by this concept in works on quantum theory, but what is, as a rule, not expressly stated. It is not difficult to note that, unlike ordinary functions, singular or improper functions are defined not by specifying their values for all values of the arguments (for some set of values of the arguments they may be infinite or not defined at all), but by specifying the rules for integrating their product with sufficiently regular functions.

In other words, an improper function is defined by specifying the corresponding linear functional in a suitable “linear space” of sufficiently regular functions.

Of course, we cannot enter here into any complete exposition of the general theory of improper functions and therefore shall confine ourselves only to the basic formulations concerning those properties of the special coefficient functions which we shall need directly below.

Let us note only that in the theory of improper integrable functions it is proved that they always possess Fourier transforms and that they can always be differentiated without violating the property of integrability.

Let us proceed to the formulation of the definition of the coefficient functions \(K(x_1,\ldots,x_n)\). For this purpose we introduce the class \(C(q,r,n)\) of functions \(F(x_1,\ldots,x_n)\), continuous together with all their derivatives up to order \(q\) inclusive, for which all products

\[ x_{j_1}^{\alpha_1}\cdots x_{j_s}^{\alpha_s}\, \frac{\partial^p F(x_1,\ldots,x_n)} {\partial x_{j_1}^{\beta_1}\cdots \partial x_{j_p}^{\beta_p}}, \qquad \begin{gathered} s=0,1,\ldots,r,\\ p=0,1,\ldots,q,\\ \alpha,\beta=0,1,2,3, \end{gathered} \tag{2.16} \]

are bounded.

In accordance with the adopted point of view, we shall regard the function \(K(x_1,\ldots,x_n)\) as given and integrable if the linear functional

\[ \int K(x_1,\ldots,x_n)F(x_1,\ldots,x_n)\,dx_1\cdots dx_n \tag{2.17} \]

is defined in the linear space of functions \(F(x_1,\ldots,x_n)\) from the class \(C(q,r,n)\), at least for sufficiently large \(q\) and \(r\). We shall refer to this requirement as the condition of integrability.

Physically, coefficient functions constructed by means of an improper limiting passage are convenient.

In the proper and improper sense we shall here call such a sequence

\[ K_M(x_1,\ldots,x_n), \qquad M\to\infty, \]

for which the corresponding sequence of integrals

\[ \int K_M(x_1,\ldots,x_n)F(x_1,\ldots,x_n)\,dx_1\cdots dx_n \tag{2.18} \]

converges in the ordinary sense for every function \(F\) from some class \(C(q,r,n)\) entering into the linear space on which the functional (2.18) is defined.

It should be emphasized that an improper limiting passage is in fact constantly used in considering singular functions in quantum theory, although, as a rule, no attention is paid to its difference from an ordinary limiting passage. This applies, for example, to all limiting definitions of the Dirac \(\delta\)-function.

Let us now consider from this point of view the question of approximating the function \(\Delta_{\alpha\beta}^{(-)}(x)\), defined by the formal relation

\[ \Delta_{\alpha\beta}^{(-)}(x) = P_{\alpha\beta}\!\left(-i\frac{\partial}{\partial x}\right)D^{(-)}(x) = \]

\[ = \frac{1}{(2\pi)^3 i} \int e^{-ikx}P_{\alpha\beta}(-k)\,\theta(k^0)\delta(k^2-m^2)\,dk, \tag{2.19} \]

where

\[ \theta(k^0)= \begin{cases} +1, & (k^0>0),\\ 0, & (k^0<0). \end{cases} \]

Let us first establish what meaning should be assigned to the integral with infinite limits that figures here. As is known, an ordinary integral taken over an infinite domain of integration is defined as the limit of integrals over finite domains of integration, expanding without bound and in the limit covering the entire given infinite domain. Naturally, therefore, integrals of the type

\[ K(x)=\int e^{-ikx}P(k)\theta(k^0)\delta(k^2-m^2)\,dk \tag{2.20} \]

with polynomial \(P(k)\) should be defined as the improper limit of the integral

\[ K_G(x)=\int_G e^{-ikx}P(k)\theta(k^0)\delta(k^2-m^2)\,dk \tag{2.21} \]

under an unbounded expansion of the domain \(G\).

Passing to the momentum representation and using the condition of integrability, one can show that such an improper limit indeed exists and does not depend on the form of the sequence of domains \(G_n\) expanding to \(G\). In this case it turns out that if the degree of \(P(k)\) is \(\nu\), then for any \(F(x)\) of class \(C(\nu+3,5,1)\) this limit has the form

\[ \int K(x)F(x)\,dx = \int \widetilde{F}(k^0,\mathbf{k})P(k^0,\mathbf{k})\,\frac{d\mathbf{k}}{2k^0}; \qquad k^0=\sqrt{\mathbf{k}^2+m^2}, \tag{2.22} \]

where

\[ \widetilde{F}(k^0,\mathbf{k}) = \widetilde{F}(k) = \int F(x)e^{-ikx}\,dx . \tag{2.23} \]

Thus, relation (2.19) indeed defines an integrable improper function \(\Delta_{\alpha\beta}^{(-)}(x)\), and

\[ \int \Delta_{\alpha\beta}^{(-)}(x)F(x)\,dx = -\frac{i}{(2\pi)^3} \int \widetilde{F}(k^0,\mathbf{k})P_{\alpha\beta}(-k^0,-\mathbf{k}) \,\frac{d\mathbf{k}}{2k^0}. \tag{2.24} \]

As has just been established, \(\Delta_{\alpha\beta}^{(-)}(x)\) is the improper limit of a sequence of regular analytic functions

\[ \frac{i}{(2\pi)^3} \int_{G_n} e^{-ikx}P_{\alpha\beta}(-k)\theta(k^0)\delta(k^2-m^2)\,dk . \tag{2.25} \]

Such an approximation, however, suffers from one drawback. The point is that expressions (2.25) are not covariant, since under Lorentz transformations the domain of integration changes.

To approximate \(\Delta_{a\beta}^{(-)}(x)\), understood in an improper sense, by continuous and at the same time covariant functions, we shall use the well-known Pauli–Villars regularization method\(^4\) with the aid of auxiliary masses, putting

\[ \left. \begin{aligned} \operatorname{reg}\bigl[\Delta_{a\beta}^{(-)}(x)\bigr] &= P_{a\beta}\left(-i\frac{\partial}{\partial x}\right) \operatorname{reg}\bigl[D^{(-)}(x)\bigr], \\[4pt] \operatorname{reg}\bigl[D^{(-)}(x)\bigr] &= D^{(-)}(x)+\sum_{1\le j<l} c_j D_{M_j}^{(-)}(x), \end{aligned} \right\} \tag{2.26} \]

where \(D_{M_j}^{(-)}(x)\) is the negative-frequency part of the Pauli function with mass \(M_j\), and the numbers \(c_j\) are determined from the system of equations

\[ \left. \begin{aligned} 1+\sum_j c_j &=0,\\ m^2+\sum_j c_j M_j^2 &=0, \\ &\cdots \\ m^{2(l-1)}+\sum_j c_j M_j^{2(l-1)} &=0. \end{aligned} \right\} \tag{2.27} \]

As can be shown, in the limit as the values of the auxiliary masses increase, the functions \(D_{M_j}^{(-)}\) tend to zero everywhere except for an infinitely small neighborhood of the light cones, and the masses \(M_j\) can always be chosen so that, as \(M_j\to\infty\), all coefficients remain bounded. Therefore one may put

\[ D^{(-)}(x)=\lim_{M_j\to\infty}\operatorname{reg}\bigl[D^{(-)}(x)\bigr], \tag{2.28} \]

where the limiting transition sign has here the improper meaning.

As follows from the structure of the \(D^{(-)}\)-function,\(^4\) the function \(\operatorname{reg}\{D^{(-)}(x)\}\) thus defined is continuous and has continuous partial derivatives up to order \((l-2)\) inclusive. Therefore, for \(l>\nu+2\) (\(\nu\) is the degree of the polynomial \(P_{a\beta}\)), the function \(\operatorname{reg}[\Delta_{a\beta}^{(-)}]\) will also be continuous.

Let us now pass to the study of the question of defining the products occurring in Wick’s theorem

\[ \prod_{(r<s)} \Delta_{a\beta}^{(-)}(x_r-x_s),\qquad r,s=1,\ldots,n. \tag{2.29} \]

Let us emphasize that the necessity of a special definition of products is typical for improper functions. The point is that

an improper function is specified by establishing rules for its integration only with sufficiently regular functions, and from such rules there does not directly follow a recipe for integrating the product of several singular functions.

Therefore we shall have to use the method of improper limiting transition and define expression (2.29) by means of a convergent sequence of regular functions.

Carrying out in (2.29) the formal multiplication with the aid of representation (2.19), we find:

\[ \prod_{(r<s)} \Delta^{(-)}_{\alpha_r \beta_s}(x_r-x_s) = \int e^{\,i\sum k_r x_r}\, \Delta(k_1,\ldots,k_n)\,dk_1\ldots dk_n, \tag{2.30} \]

where

\[ \Delta(k_1,\ldots,k_n) = \frac{i^N}{(2\pi)^{3N}} \int \prod_r \delta\!\left( k_r+\sum_{s<r}\lambda_{\alpha_s\beta_r} -\sum_{r<s}\lambda_{\alpha_r\beta_s} \right) \times \prod_{r<s} \left\{ P_{\alpha\beta}(-\lambda_{\alpha\beta}) \theta(\lambda^0_{\alpha\beta}) \delta(\lambda^2_{\alpha\beta}-m^2_{\alpha\beta}) \,d\lambda_{\alpha\beta} \right\}, \tag{2.31} \]

and \(N\) is the number of factors in the product (2.29). It is not difficult to verify that the domain of integration in (2.31) is in fact finite. This is based on the circumstance that from the condition of boundedness of the sum of positive frequencies \(\lambda^0_{\alpha\beta}\)

\[ \sum_{s>r}\lambda^0_{r\beta} = k^0_r+\sum_{s<r}\lambda^0_{\alpha_s\beta_r} \]

there follows the boundedness of each individual frequency \(\lambda^0_{\alpha\beta}\), and consequently also of the momentum

\[ \bar{\lambda}_{\alpha\beta} = \sqrt{(\lambda^0_{\alpha\beta})^2-m^2}. \]

The indicated finiteness of the domain of integration in (2.31) implies the absence of divergences and the possibility of a rigorous definition of expression (2.30) as the improper limit of a sequence of regular analytic functions

\[ \lim_{\Gamma\to\infty} K(x_1,\ldots,x_n/\Gamma)= \]

\[ = \lim_{\Gamma\to\infty} \int_{\Gamma} e^{\,i\sum k_r x_r}\, \Delta(k_1,\ldots,k_n)\,dk_1\ldots dk_n \tag{2.32} \]

under an unbounded expansion of the domain \(\Gamma\) of the four-dimensional space of the points \(k_1,\ldots,k_n\), in the limit covering this entire space.

In this case it turns out that the corresponding limiting equality of functionals

\[ \lim_{I\to\infty}\int K(x_1,\ldots,x_n\mid I)F(x_1,\ldots,x_n)\,dx_1\ldots dx_n = \]

\[ =\int F(x_1,\ldots,x_n)\prod_{(r<s)}\Delta_{\alpha_r\beta_s}^{(-)}(x_r-x_s)\,dx_1\ldots dx_n \tag{2.33} \]

holds in the class \(C(\nu+2N+1,\;4n+1,\;n)\), where \(\nu\) is the sum of the degrees of all \(N\) polynomials \(P_{\alpha\beta}\).

The products (2.29) of singular functions \(D_{\alpha\beta}^{(-)}\) under consideration may also be approximated with the aid of the corresponding products of regularized covariant functions \(\operatorname{reg}\{D_{\alpha\beta}^{(-)}\}\). In this case relation (2.33) takes the form

\[ \lim_{\alpha_j\to\infty}\int F(x_1,\ldots,x_n) \left\{\prod_{r<s}\operatorname{reg}\Delta_{\alpha_r\beta_s}^{(-)}(x_r-x_s)\right\} \,dx_1\ldots dx_n = \]

\[ =\int F(x_1,\ldots,x_n) \left\{\prod_{r<s}\Delta_{\alpha_r\beta_s}^{(-)}(x_r-x_s)\right\} \,dx_1\ldots dx_n \tag{2.34} \]

for \(F(x_1,\ldots,x_n)\in C(\nu+2N+1,\;4n+1,\;n)\).

Let us also note that the arguments given above extend also to the general case of coefficient functions obtained as a result of multiplying two operator functions of type (2.1) with different arguments, and the corresponding expressions

\[ K(x_1,\ldots,x_n)Q(y_1,\ldots,y_m) \prod \Delta_{\alpha\beta}^{(-)}(x_i-y_j) \]

(in which \(x_1,\ldots,x_n,\;y_1,\ldots,y_m\) are independent arguments, while \(K\) and \(Q\) are integrable, translationally invariant coefficient functions) may be defined as integrable improper functions.

For expressions of this kind one can prove the following general theorem: if \(K_M(x_1,\ldots,x_n)\), \(Q_M(y_1,\ldots,y_m)\) are translationally invariant coefficient functions and, in the improper sense,

\[ K_M(x_1,\ldots,x_n)\xrightarrow[M\to\infty]{}K(x_1,\ldots,x_n); \qquad Q_M(y_1,\ldots,y_m)\xrightarrow[M\to\infty]{}Q(y_1,\ldots,y_m), \]

and the arguments \(x_1,\ldots,x_n,\;y_1,\ldots,y_m\) are independent, then

\[ K_M(x_1,\ldots,x_n)Q_M(y_1,\ldots,y_m) \prod\left(\operatorname{reg}\Delta_{\alpha\beta}^{(-)}(x_i-y_j)\right)\to \]

\[ \to K(x_1,\ldots,x_n)Q(y_1,\ldots,y_m) \prod\left(\Delta_{\alpha\beta}^{(-)}(x_i-y_j)\right). \tag{2.35} \]

The proof of this theorem, like the proof of relation (2.34), is based on the simple circumstance that from

boundedness of the sum of negative frequencies also implies the boundedness of each individual frequency, and is due to the fact that we are considering products of only negative-frequency parts of the $\Delta$-function. Our analysis is therefore trivially carried over also to the case when, instead of $\Delta^{(-)}$, there stand only $\Delta^{(+)}$-functions.

In the theory of interacting fields one has to deal with products of the so-called causal functions $\Delta^c_{\alpha\beta}(x)$:

\[ \Delta^c_{\alpha\beta}(x) = P_{\alpha\beta}\left(-i\frac{\partial}{\partial x}\right)D^c(x), \]

\[ D^c(x) = \frac{1}{(2\pi)^4} \int \frac{e^{ikx}\,dk}{m^2-k^2-i\varepsilon}, \]

\[ D^c(x) = \begin{cases} D^{(-)}(x), & \text{for } x^0>0,\\ -D^{(+)}(x), & \text{for } x^0<0, \end{cases} \]

or, in other words, Feynman propagation factors.

As is seen, both positive and negative frequencies are present in the $\Delta^c$-functions. In view of this, a product of the type

\[ \prod_{(r<s)} \Delta^c_{\alpha_r\alpha_s}(x_r-x_s) \]

is no longer a definite integrable function, and, strictly speaking, a limiting relation of the type (2.34) has no place here.

Let us now turn to the consideration of operator expressions represented by finite sums of the form (2.1). Let us first note that the existence of the limiting relations (2.34) removes one possible objection to the proof of Wick’s theorem given above, based on the fact that in the course of this proof we performed multiplication of singular functions.

A proof of Wick’s theorem that is completely free of this shortcoming is carried out very simply, for example, according to the following scheme.

We first consider the fictitious case in which the field operators satisfy commutation relations in which $\Delta_{\alpha\beta}$ is replaced by $\operatorname{reg}\{\Delta_{\alpha\beta}\}$, and for it we carry out the proof of Wick’s theorem in exactly the form in which it was set forth above. In view of the continuity of the regularized functions, such a proof will already be quite rigorous. To pass to the real case it is now sufficient to perform the limiting transition $M\to\infty$, which removes the regularization.

In the same way, with the aid of the more general limiting relations (2.35), it is not difficult to justify the legality of multiplying two integrable operator functions

\[ A_1(x_1,\ldots,x_n)A_2(y_1,\ldots,y_m) \]

with independent arguments.

We shall call integrable those operator expressions represented by finite sums of the type (2.1), in which all coefficient functions satisfy the integrability requirement.

Let us now take some integrable operator function \(A(x_1,\ldots,x_n)\) and consider the integral

\[ \int A(x_1,\ldots,x_n)F(x_1,\ldots,x_n)\,dx_1\cdots dx_n . \tag{2.36} \]

Passing to the matrix elements of expression (2.36), by means of formulas (2.5) we find that these elements will contain the integrals

\[ \int K_{\ldots\alpha\ldots\beta\ldots}(x_1,\ldots,x_n) e^{i\sum p_\nu x_\nu-i\sum p'_\lambda x_\lambda} \times F(x_1,\ldots,x_n)\,dx_1\cdots dx_n, \]

which, on the basis of the integrability property of the coefficient functions \(K_{\ldots\alpha\ldots\beta\ldots}\), turn out to be finite. The operator integrals (2.36) may therefore be regarded as convergent.

In the same sense, integrals of the type

\[ \int A(x,x_1,\ldots,x_n)F(x_1,\ldots,x_n)\,dx_1\cdots dx_n \tag{2.37} \]

with one integration not carried out will also be convergent. This fact follows directly from the property of translational invariance of the function \(A(x_1,\ldots,x_n)\).

Having established the convergence of operator integrals of the form (2.36) and (2.37), let us note that, because of the generality of our integrability condition, the class of operator integrals with guaranteed convergence proves to be somewhat narrow. Thus, this class does not include the integrals

\[ \int A(x_1,\ldots,x_n)\,dx_1\cdots dx_n, \]

taken over the whole infinite space of points \((x_1,\ldots,x_n)\), since they correspond to the function \(F=1\), which does not decrease at infinity. The matrix elements of operator integrals of this kind therefore require a special definition.

Up to now we have considered operator functions of the points \(x_1,\ldots,x_n\) expressed in terms of positive- and negative-frequency parts of quantized field functions.

Of great importance is a more special type of operator functions, into which the quantized field functions enter, so to speak, “as a whole,” under the sign of normal products

\[ A(x_1,\ldots,x_n)=\sum K_{\ldots\alpha\ldots}(x_1,\ldots,x_n):\ldots u_\alpha(x_j)\ldots: . \tag{2.38} \]

If the fermion field functions enter here only in even combinations, then such a sum will be called a polylocal operator.

From this definition and Wick’s theorem it follows at once that the multiplication of several polylocal operators again leads to a polylocal operator.

Polylocal operators possess the following important property:

\[ [A_1(x_1,\ldots,x_n);\ A_2(y_1,\ldots,y_m)]_- = 0, \tag{2.39} \]

if each of the \(x_i\) is space-like with respect to each of the \(y_j\). This property follows directly from the vanishing of the permutation functions \(\Delta_{\alpha\beta}(x_i-y_j)\) outside the light cone and from the even number of anticommuting fermion functions in the polylocal operators \(A_1\) and \(A_2\).

Let us now consider the case when \(n=1\). Then the polylocal operator depends only on the behavior of the field functions at a single point \(x\), in view of which we shall call it a local operator. From (2.39), for two local operators it follows that

\[ [A_1(x),\ A_2(y)]_- = 0 \tag{2.40} \]

when \(x\) is space-like with respect to \(y\).

Let us also note that, for \(n>1\), it may turn out that a polylocal operator in fact depends on the behavior of the field functions only at one point.

Suppose, in fact, that in the sum (2.38) all coefficient functions \(K\) vanish for all \(x_1,\ldots,x_n\), except those satisfying the equality

\[ x_1=x_2=\cdots=x_n. \]

It is clear that such \(K\) can be constructed only from the expression

\[ \delta(x_1-x_2)\cdots\delta(x_1-x_n) \]

and its partial derivatives. Since the coefficient functions, by definition, must possess the property of translational invariance, the general expression for them in the case under consideration has the form

\[ Z\left(\cdots \frac{\partial}{\partial x}\cdots\right)\delta(x_1-x_2)\cdots\delta(x_1-x_n), \]

where \(Z\left(\cdots \frac{\partial}{\partial x}\cdots\right)\) is a certain polynomial in \(\dfrac{\partial}{\partial x_i^a}\) with constant coefficients.

A polylocal operator with coefficient functions of this type will be called quasilocal.

Integration of a quasilocal operator over all points \(x_1,\ldots,x_n\), except one, leads to an ordinary local operator.

§ 3. BASIC CONCEPTS OF THE THEORY OF INTERACTING FIELDS

One of the most widespread methods used in ordinary quantum mechanics for investigating the behavior of dynamical systems is the well-known Schrödinger representation. In this representation the behavior of a dynamical system is described by means of a time-dependent wave function \(\Psi(t)\), determined by the Schrödinger equation

\[ i\,\frac{\partial \Psi(t)}{dt}=H\Psi . \tag{3.1} \]

Here \(H\) is the Hamiltonian operator, corresponding to the total energy of the entire system and not depending on time for closed systems. In general, in this representation the dynamical variables of closed systems are characterized by operators that do not depend explicitly on time. Their mean values

\[ \overline{A}_{t}=(\Psi^{*}(t)A\Psi(t)) \tag{3.2} \]

may, however, depend on time through the wave functions \(\Psi(t)\). Regarding the operator \(H\) as constant in time, we can formally integrate equation (3.1) and write

\[ \Psi(t)=e^{\frac{Ht}{i}}\Psi, \]

where \(\Psi=\mathrm{const}\) does not depend on time.

Substituting this expression into (3.2), we obtain:

\[ \overline{A}_{t}=\left(\Psi^{*}e^{-\frac{Ht}{i}}Ae^{\frac{Ht}{i}}\Psi\right). \tag{3.3} \]

Formula (3.3) can be interpreted as the mean, with respect to time-independent functions \(\Psi\), of the operator

\[ A(t)=e^{-\frac{Ht}{i}}Ae^{\frac{Ht}{i}}, \tag{3.4} \]

which depends on time. We thus arrive at the Heisenberg representation, in which it is not the wave functions that explicitly depend on time, but the dynamical variables. As is evident, from the point of view of computing the observed values of dynamical quantities, the two representations are completely equivalent. Differentiating (3.4) with respect to time, we see that in the Heisenberg representation the law of evolution of dynamical variables is determined by the equation

\[ i\,\frac{\partial A(t)}{dt}=[A(t),H]. \tag{3.5} \]

where

\[ [a,b]_{-}=ab-ba \]

are the quantum Poisson brackets.

It is interesting to note that in classical mechanics the course of the time dependence of dynamical quantities is determined by the same equations (3.5), with the classical Poisson brackets.

When applying the Schrödinger representation to the case of interacting quantized fields that interests us, one has to divide the complete Hamiltonian \(H\) into two parts,

\[ H=H_0+H_1, \tag{3.6} \]

where \(H_0\) is the Hamiltonian of the free field, and \(H_1\) is the Hamiltonian of the interaction:

\[ H_0=\int H_0(x)\,dx;\qquad H_1=\int H_1(x)\,dx. \tag{3.7} \]

Here the energy densities \(H_0(x)\) and \(H_1(x)\) are combinations of the fundamental field functions referred to some fixed instant of time, for example,

\[ u(0,\mathbf{x})=u(x)\big|_{x^0=0}. \]

Therefore, when considering processes of interaction of fields, one usually starts from the Schrödinger equation written in the form

\[ i\,\frac{\partial\Psi(t)}{\partial t}=(H_0+H_1)\Psi(t). \tag{3.8} \]

One of the very fundamental shortcomings of the contemporary quantum theory of fields of real interacting particles is the necessity of introducing into consideration fields of fictitious noninteracting particles and treating the interaction as a given additional factor, changing the properties of the dynamical system, a factor which can be “switched on” and “switched off.” At first glance there would seem to be no special grounds for criticism here. Indeed, particles are in intense interaction with one another only when they are sufficiently close. It would therefore seem that, at large distances between them, the interaction of the fields is insignificant, and in some approximation it is expedient to abstract from it and to regard the real particles as free.

Reasoning in this way, however, we lose sight of the fact that particles constantly interact with the vacuum as with a kind of physical “medium” in which they move.

It therefore seems desirable that, in constructing the theory, one should from the very beginning deal with real interacting particles and not introduce the artificial representation of fictitious free fields.

Since, however, up to the present time no such program has been carried out, one must resort to such a division.

In the absence of interaction, the basic equation (3.8) takes the form

\[ i\frac{\partial \Psi(t)}{\partial t}=H_0\Psi(t). \]

Carrying out a formal integration, we find:

\[ \Psi(t)=e^{\frac{H_0t}{i}}\Phi . \tag{3.9} \]

The constant \(\Phi\) in the study of free fields was called by us the amplitude of the state. In the case of interacting fields, expression (3.9) will no longer satisfy the basic equation with \(\Phi\) independent of time. However, we can generalize it by putting

\[ \Psi(t)=e^{\frac{H_0t}{i}}\Phi(t), \tag{3.10} \]

and regard \(\Phi(t)\) as a new amplitude of the state. From the mathematical point of view, we are here making a replacement of the unknown function by a method analogous to the method of variation of an arbitrary constant used in the theory of differential equations. Substituting (3.10) into the basic equation (3.8), we obtain:

\[ H_0e^{\frac{H_0t}{i}}\Phi(t)+ie^{\frac{H_0t}{i}}\frac{\partial\Phi(t)}{\partial t} =(H_0+H_1)e^{\frac{H_0t}{i}}\Phi(t), \]

whence

\[ i\frac{\partial\Phi(t)}{\partial t} =e^{-\frac{H_0t}{i}}H_1e^{\frac{H_0t}{i}}\Phi(t). \]

Let us discuss the meaning of the equation obtained. In equation (3.7) the interaction-energy density \(H_1(x)\) is a linear combination of expressions of the type

\[ H_j(x)=u_\alpha(x,0)\ldots u_\gamma(x,0), \]

but

\[ e^{-\frac{H_0t}{i}}H_je^{\frac{H_0t}{i}} = e^{-\frac{H_0t}{i}}u_\alpha(x,0)e^{\frac{H_0t}{i}} \ldots e^{-\frac{H_0t}{i}}u_\gamma(x,0)e^{\frac{H_0t}{i}} . \]

Therefore the action of the operator

\[ e^{-\frac{H_0t}{i}}\ldots e^{\frac{H_0t}{i}} \]

on \(H_1\) corresponds to replacing the operator field functions \(u(x,0)\),

independent of time, into time-dependent operator functions

\[ u(x)=u(x,t)=e^{-\frac{H_0 t}{i}}\,u(x,0)\,e^{\frac{H_0 t}{i}} . \]

We thus pass here to the Heisenberg representation for the operator functions of the free field.

To determine \(\Phi(t)\) we now have an equation of the form:

\[ i\frac{\partial\Phi}{\partial t}=H_1(t)\Phi(t), \]

\[ H_1(t)=\int H_1(x,t)\,dx, \tag{3.11} \]

where the operator \(H_1(x,t)\) is obtained from \(H_1(x)\) by replacing \(u(x,0)\) by \(u(x)\). Such a representation of the Schrödinger equation is called the interaction representation.

Let us consider the question of the expression of dynamical variables in the interaction representation. Substituting (3.10) into (3.2), we obtain:

\[ \bigl(\overset{*}{\Psi}(t)A\Psi(t)\bigr) = \left( \overset{*}{\Phi}(t)e^{-\frac{H_0 t}{i}} Ae^{\frac{H_0 t}{i}}\Phi(t) \right), \]

i.e.,

\[ \overline{A_t}= \bigl(\overset{*}{\Phi}(t)A_{\mathrm{int}}(t)\Phi(t)\bigr), \]

where

\[ A_{\mathrm{int}}(t)=e^{-\frac{H_0 t}{i}}Ae^{\frac{H_0 t}{i}} . \]

Thus we see that, in the interaction representation, operator expressions of dynamical variables must be regarded as functions of the field operators in the Heisenberg representation for free fields. In other words, the operators of dynamical variables are expressed by forms depending on the field functions

\[ u(x,t)=u(x), \]

satisfying the homogeneous equations of the free fields.

As is evident, the main shortcoming of all the indicated representations of the Schrödinger equation, including the interaction representation, is the distinguished role of time and, consequently, the manifest noncovariance of the formulation. This formal shortcoming of the theory was removed in a special modification of the interaction representation developed by Tomonaga and Schwinger\(^{5,6}\). In the indicated modification, instead of working with the surfaces \(t=\mathrm{const}\) of four-dimensional space-time, a more general class of space-like surfaces \(\sigma\) is introduced.

Let us note that already with the aid of equation (3.11) we can introduce into the discussion a very important characteristic of the system, the so-called scattering matrix, or \(S\)-matrix. Suppose that we are studying a process at the beginning and at the end of which there are only particles far removed from one another, which may be regarded as free.

To calculate the probability amplitude for the scattering and mutual transformations of particles occurring in this process, let us consider the situation in which the interaction \(H_1(t)\) is switched on adiabatically in the infinitely remote past and switched off adiabatically in the infinitely remote future. Then, denoting the amplitude of the initial state by \(\Phi(-\infty)\), and the amplitude of the final state by \(\Phi(\infty)\), we can relate them by the relation

\[ \Phi(\infty)=S\Phi(-\infty), \tag{3.12} \]

in which the operator \(S\) is called the scattering operator or the scattering matrix. The squares of the corresponding matrix elements of the operator \(S\) determine the transition probabilities and effective cross sections of possible scattering processes and mutual transformations of particles.

To obtain computational formulas, one could, starting from equation (3.11), construct its solution by the method of successive approximations in the form of an expansion in powers of the smallness of the interaction. We would then obtain a relation between \(\Phi(-\infty)\) and \(\Phi(\infty)\) of the form (3.12), where the operator \(S\) is written as the expansion

\[ S=1-i\int_{-\infty}^{\infty} H_1(t)\,dt +(-i)^2\int_{-\infty}^{+\infty} H_1(t)\,dt \int_{-\infty}^{t} H_1(t')\,dt' + \cdots . \]

It is precisely in this way, i.e. by starting from the Schrödinger equation in some variant of the interaction representation, that the \(S\)-matrix is investigated in the majority of existing studies in quantum field theory\(^{2,6,7}\).

In our opinion, it is more expedient to proceed from the scheme proposed by Stueckelberg\(^{8,9}\), in which a generalized \(S\)-matrix is introduced without recourse to the Hamiltonian formalism. Its role in specifying the form of the \(S\)-matrix is replaced by explicitly formulated physical conditions, among which the causality condition plays an important role.

Stueckelberg, however, did not succeed in obtaining a sufficiently clear and general formulation of such a condition, as a result of which his ideas did not find broad application.

In the direction of a further development of these ideas, we shall attempt below to develop a new formulation of the causality condition

and the method, based on it, for constructing the quantum theory of interacting fields*).

In constructing the theory, as in the usual exposition, we shall have to use the operations of “switching on” and “switching off” the interaction.

In order to describe this operation mathematically, introduce a function \(g(x)\) with values in the interval \((0,1)\), representing the degree of intensity with which the interaction is switched on. Then in those regions where \(g(x)=0\) the interaction is absent; in those regions where \(g(x)=1\) it is switched on completely, and for \(0<g(x)<1\) the interaction is switched on only partially.

Replacing the actual interaction Lagrangian \(L(x)\) by the product \(L(x)g(x)\), we obtain a situation in which the interaction is “switched on with intensity \(g(x)\).”

Now let \(g(x)\) differ from zero only in some finite space-time region. In this case, in the sufficiently distant past and future the fields are free, and therefore the initial and final states of the dynamical system can be characterized by the usual constant state amplitudes.

These two state amplitudes \(\Phi(-\infty)\) and \(\Phi(\infty)\) will be related by some operator \(S(g)\), transforming \(\Phi(-\infty)\) into \(\Phi(\infty)\) and depending on the behavior of the function \(g(x)\).

Fixing the initial state amplitude \(\Phi(-\infty)=\Phi\), we may regard the final amplitude as a functional of \(g\)

\[ \Phi(g)=S(g)\Phi . \tag{3.13} \]

By this definition, \(S(g)\) is naturally interpreted as the scattering matrix for the case when the interaction is switched on with intensity \(g(x)\). The real case, when the interaction is fully switched on throughout all space-time, must in this scheme be considered by means of a limiting transition, in which the region where \(g(x)=1\) is enlarged without bound and in the limit covers all of space-time. If in this case, at least for some matrix elements of \(S(g)\), limiting values exist, then these limiting values may be regarded as the corresponding elements of the usual scattering matrix, which can be written formally as

\[ S=S(1). \]

Let us now relate the operator \(S(g)\) to the interaction Lagrangian. As is known, in the classical theory the interaction is taken into account

*) Reported by N. N. Bogoliubov at a meeting of the Physico-Mathematical Division of the Jubilee Session of the Academy of Sciences of the USSR in April 1954.

by adding to the Lagrangian of the free field \(L_0(x)\) the interaction Lagrangian \(L(x)\), and the equations of motion may be obtained with the aid of the principle of stationary action.

Let us consider the action \(A\) in a system of classical fields in the case when the interaction is included with intensity \(g(x)\). We then have:

\[ A=\int \{L_0(x)+g(x)L(x)\}\,dx, \tag{3.14} \]

where in (3.14) under the integral sign there stand field functions satisfying the corresponding equations of motion. In particular, if \(g(x)\) is regarded as an infinitesimal quantity of the first order, then these field functions will differ from the free field functions also by infinitesimal quantities of the first order. On the other hand, in view of the fact that the equations of the free fields are obtained from the principle

\[ \delta \int L_0(x)\,dx=0, \]

it follows that if, under the integral \(\int L_0\,dx\), the field functions are written with accuracy up to infinitesimal quantities of the first order, then this will cause an error of second order of smallness in the value of the integral.

Therefore, when the interaction is switched on with infinitesimal intensity \(g(x)\), the action in the system changes by the amount

\[ \int L(x)g(x)\,dx, \tag{3.15} \]

in which \(L(x)\) depends on the wave functions of the free field.

As is known\(^*\), in the limiting quasiclassical case the solution of the ordinary Schrödinger equation for the wave function \(\psi(t)\)

\[ i\frac{\partial \psi}{\partial t}=H\psi \]

takes the form

\[ \psi=e^{iA}, \]

where \(A\) is the action of the system. To the transition from the unperturbed expression

\[ A_0=\int L_0(x)\,dx \tag{3.16} \]

to the action (3.14) there will, obviously, correspond the transformation of the wave function

\[ \psi_0=e^{iA_0}\longrightarrow \psi=e^{iA}=e^{i\int L(x)g(x)\,dx}\psi_0. \]

\(^*\) See, for example, \(^{11,12}\).

Thus, taking into account the smallness of the quantity \(g'(x)\), the transition from (3.16) to (3.14) corresponds to an infinitesimal transformation of the wave function

\[ \psi \to \psi'=\psi+\delta\psi;\quad \delta\psi=i\int L(x)g(x)\,dx\,\psi. \]

Starting from the correspondence principle, we shall require that the transformation law of the second-quantized state amplitude \(\Phi\) have the same form, i.e.

\[ \Phi \to \Phi'=\Phi+\delta\Phi;\quad \delta\Phi=i\int L(x)g(x)\,dx\,\Phi. \]

In other words, we shall assume that for infinitesimal \(g(x)\) the matrix \(S(g)\) has the form

\[ S(g)=1+i\int L(x)g(x)\,dx. \tag{3.17} \]

In order to determine \(S(g)\) in terms of the interaction Lagrangian \(L(x)\) in the real case of interest to us, when \(g(x)\) is not only not small, but is close to unity, it is necessary to formulate the remaining conditions that \(S(g)\) must satisfy.

The basic physical requirement, as always, is the requirement of relativistic covariance. To formulate this requirement, let us consider a transformation \(L\) from the extended Lorentz group (including translations of the coordinate system)

\[ x\to Lx. \tag{3.18} \]

In the absence of interaction, in the theory of the free field, the transformation law of the state amplitude corresponding to (3.18) had the form

\[ \Phi'=U_L\Phi. \tag{1.1} \]

In the case under consideration, when the state amplitude \(\Phi(g)\) depends on the function \(g\), it is necessary to take into account that the function \(g(x)\) itself, which may be regarded as a certain “classical field,” under the transformation (3.18) undergoes a transformation of the form

\[ g(x)\to Lg(x)=g(Lx). \]

Therefore the transformation law of the amplitude \(\Phi(g)\) will be

\[ \Phi'(Lg)=U_L\Phi(g). \tag{3.19} \]

From the consideration of relativistic covariance it is also necessary to require that the transformation law (3.13) from the initial function to the final one not depend on the reference frame, i.e.

\[ \Phi'(g)=S(g)\Phi'. \tag{3.20} \]

Substituting here relation (1.1) and the shifted \((Lg-\varkappa)\) relation (3.19), we find, taking (3.13) into account,

\[ U_L S(L^{-1}g)\Phi = S(g)U_L\Phi . \]

In view of the arbitrariness of the amplitude of the initial state \(\Phi\), this expression can be written in operator form as

\[ U_L S(L^{-1}g)=S(g)U_L \]

or, shifting the arguments by \(L\) and multiplying on the right by \(U_L^{-1}\), taking into account the unitarity of \(U_L\),

\[ S(Lg)=U_L S(g)U_L^{-1}=U_L S(g)U_L^{*}. \tag{3.21} \]

Formula (3.21) expresses the covariance condition for the operator \(S(g)\).

Let us now formulate another general requirement imposed in quantum mechanics on the laws governing the change of wave functions, namely the requirement that their norm be preserved. Applied to the present case, we must require

\[ (\Phi^{*}(g)\Phi(g))=(\Phi^{*}\Phi), \]

whence it follows that

\[ (\Phi^{*}S^{*}(g)S(g)\Phi)=(\Phi^{*}\Phi), \]

i.e. the operator \(S(g)\) must be unitary,

\[ S^{*}(g)S(g)=1. \tag{3.22} \]

We must also ensure the fulfillment of the causality condition, according to which any event that has occurred in the system can influence the course of its evolution only in the future and cannot influence the behavior of the system in the past, at times preceding the given event.

We must therefore require that a change in the law of interaction in some space-time region be able to influence the evolution of the system only at subsequent moments of time.

In order to formulate this condition, let us fix in some reference system a certain time \(t\) and require that the unitary operator \(S(g)\), which transforms the initial amplitude \(\Phi(-\infty)\) into the final one \(\Phi(\infty)=\Phi(g)\), can be represented as the product of two unitary operators

\[ S(g)=S_2^{t}(g)S_1^{t}(g), \tag{3.23} \]

of which the first operator \(S_1^{t}(g)\) determines the evolution of the system from \(-\infty\) up to the moment \(t\) and therefore does not depend on the behavior of the function \(g(x)\) for \(x^{0}>t\), while the second operator \(S_2^{t}(g)\) determines

QUESTIONS OF QUANTUM FIELD THEORY

the evolution of the system, beginning from the moment \(t\), and does not depend on the behavior of \(g(x)\) for \(x^0<t\).

Condition (3.23) is the expression of the causality principle in the theory of the matrix \(S(g)\).

Let us note that in the case where the region \(G\), in which the function \(g\) is different from zero, splits into two separate subregions such that all points of one of them, \(O_1\), lie in the past relative to the time \(t\), and all the other points of the other, \(O_2\), lie in the future, relation (3.23) takes the form

\[ S(g)=S(g_2)S(g_1), \]

where \(g_2\) is the part of the function \(g\) differing from zero only in \(O_2\), and \(g_1\) is the part of the function \(g\) differing from zero only in \(O_1\). In other words, in this case the matrix \(S(g)\) is equal to the product of the matrices \(S(g_2)\) and \(S(g_1)\), each of which corresponds exclusively to an interaction with intensity \(g_2\) and \(g_1\), respectively, and depends only on its own region \((O_2\) or \(O_1)\), and does not depend on the time moment \(t\).

In the more general case considered by us, the matrices \(S'_1\) and \(S'_2\) also depend on an infinitely small neighborhood of \(t\), and therefore are not scattering matrices for the corresponding regions \(x^0>t\) and \(x^0<t\).

In order to formulate the causality conditions without explicitly introducing these quantities, let us consider an infinitely small variation \(\delta g\) of the function \(g(x)\) at a point \(y\) such that

\[ y^0>t. \]

The limiting wave function \(\Phi(g)\) will then receive the increment

\[ \delta \Phi(g)=\delta S(g)\Phi=\delta S(g)\dot S(g)\Phi(g). \]

Substituting here expression (3.23), we find, taking into account the unitarity of \(S'_1\), that the operator

\[ \delta S(g)\dot S(g)=\delta S'_2(g)S'_1(g)\dot S'_1(g)\dot S'_2(g)=\delta S'_2(g)\dot S'_2(g) \tag{3.24} \]

does not depend on the behavior of the function \(g(x)\) for

\[ x^0<t<y^0. \]

Consequently, by considerations of covariance, the operator

\[ \delta S(g)\dot S(g) \]

also cannot depend on the behavior of the function \(g(x)\) when \(x\sim y\) (\(\sim\) is the sign of spacelike similarity, i.e. \(x\sim y\) denotes,

that the points \(x\) and \(y\) are separated by a space-like interval).

Making use of the well-known concept of a variational derivative, we can thus formulate the condition of causality as the condition of independence of the expression

\[ \frac{\delta S(g)}{\delta g(y)}\dot S(g) \]

from the behavior of the function \(g(x)\) at the point \(x\) for \(x<y\) (the symbol \(x<y\) expresses the fact that the point \(x\) is situated earlier in time than \(y\), or is space-like to it).

The causality condition can evidently be written in the form

\[ \frac{\delta}{\delta g(x)} \left\{\frac{\delta S(g)}{\delta g(y)}\dot S(g)\right\}=0 \quad \text{for } x<y. \tag{3.25} \]

As will be shown in the next paragraph, the indicated conditions of relativistic covariance (3.21), unitarity (3.22), and causality (3.25), together with the correspondence principle (3.17), make it possible to determine the operator \(S(g)\) completely through the interaction Lagrangian \(L(x)\).

§ 4. THE INTERACTION LAGRANGIAN AND THE \(S\)-MATRIX

We now proceed to the actual construction of the scattering matrix through the interaction Lagrangian \(L(x)\). We shall seek \(S(g)\) in the form of a formal functional expansion in powers of \(g\)

\[ S(g)=1+\sum_{n>1}\frac{1}{n!}\int S_n(x_1,\ldots,x_n)g(x_1)\cdots g(x_n)\,dx_1\cdots dx_n. \tag{4.1} \]

In it \(S_n(x_1,\ldots,x_n)\) are operator expressions depending on the complete field functions and their partial derivatives at the points \(x_1,\ldots,x_n\). To ensure the scalar character of \(S_n\), we shall also assume that the fermionic field operators enter into \(S_n\) only in even combinations. In other words, we shall require that \(S_n(x_1,\ldots,x_n)\) be polylocal operators in the sense of the definition given in § 2. It should be emphasized that the requirement that \(S_n\) depend on the field functions as a whole, and not on their individual positive- and negative-frequency parts, is a special physical condition. When it is fulfilled, the relation

\[ \left[S_n(x_1,\ldots,x_n),\,S_m(y_1,\ldots,y_m)\right]=0, \tag{4.2} \]

holds whenever all \(x_i\) are space-like with respect to all \(y_j\). Therefore, if two

the functions \(g_1(x)\) and \(g_2(x)\) are localized in space-time regions such that any point of one region is space-like with respect to all points of the other region, then \(S(g_2)\) commutes with \(S(g_1)\). This, in essence, expresses the fact that a signal cannot propagate with a speed greater than the speed of light, and the processes of switching on the interaction in mutually space-like regions \(g_1\) and \(g_2\) do not interfere with one another. Precisely in order to ensure this important physical property we shall assume that \(S_n\) is a polylocal operator.

Let us note, in particular, that since according to (3.16)

\[ S_1(x)= iL(x), \]

the interaction Lagrangian \(L(x)\) must be a local operator. This requirement is satisfied by all commonly used forms of \(L(x)\).

In order to ensure the convergence of at least the individual terms of the expansion (4.1), we shall assume that the \(S_n\) are integrable operator functions. Indeed, then for sufficiently smooth and sufficiently rapidly decreasing functions \(g(x)\) the individual integrals in (4.1) will converge. Of course, the convergence of the individual terms of the series (4.1) has no bearing on the convergence of the entire series as a whole. At present, in connection with investigations carried out recently, \(^{13,14,15,16}\) there are even rather weighty grounds for expecting that the series (4.1) written by us must turn out to be divergent. In the best case, for sufficiently weak interactions one may hope that, by taking some small number of terms in the expansion, we obtain an approximation all the more accurate the weaker the interaction. In other words, in certain cases the series (4.1) may be regarded as a source of asymptotic approximations. One such case, important in practice, is electrodynamics.

However, even for an interaction known not to be weak (for example, the meson-nucleon interaction), the study of the formal expansion (4.1) is of great interest, since in this way we shall be able to clarify, rather simply, a number of properties of the matrix \(S(g)\), both qualitative and quantitative, in order subsequently to attempt to establish them on a more rigorous basis. In our opinion, the study of this formal series therefore has a definite value as a means of heuristic investigation. Moreover, it corresponds most fully to the present factual state of the theory, where it has still not proved possible to dispense with various formal expansions in powers of the smallness of the interaction, and where all the principal results have been obtained with their aid.

Returning to expression (4.1), we see that, without loss of generality, \(S_n(x_1,\ldots,x_n)\) may be regarded as symmetric functions of their arguments \(x_1,\ldots,x_n\), since the weight functions \(g(x_1)\ldots g(x_n)\) enter symmetrically.

We shall now disclose the conditions satisfied by the matrix \(S(g)\) in order to determine the concrete form of the functions \(S_n\).

From the condition (3.21) of relativistic covariance we have:

\[ \int U_L S_n(x_1,\ldots,x_n)\dot U_L g(x_1)\ldots g(x_n)\,dx_1\ldots dx_n = \int S_n(x_1,\ldots,x_n)g(Lx_1)\ldots g(Lx_n)\,dx_1\ldots dx_n . \tag{4.3} \]

Making in the left-hand side the change of variables \(x\to Lx\), we obtain from this:

\[ \int U_L S_n(Lx_1,\ldots,Lx_n)\dot U_L g(Lx_1)\ldots g(Lx_n)\,dx_1\ldots dx_n = \]

\[ = \int S_n(x_1,\ldots,x_n)g(Lx_1)\ldots g(Lx_n)\,dx_1\ldots dx_n, \]

whence we arrive at the condition of Lorentz covariance for \(S_n\)

\[ U_L S_n(Lx_1,\ldots,Lx_n)\dot U_L=S_n(x_1,\ldots,x_n) \]

or

\[ S_n(Lx_1,\ldots,Lx_n)=\dot U_L S_n(x_1,\ldots,x_n)U_L . \tag{4.4} \]

In order to take into account the unitarity of the \(S\)-matrix (3.22), we multiply the expansion (4.1) by its adjoint

\[ \dot S(g)=1+\sum_{n>1}\frac{1}{n!}\int \dot S_n(x_1,\ldots,x_n)g(x_1)\ldots g(x_n)\,dx_1\ldots dx_n \tag{4.5} \]

and, denoting for symmetry of notation

\[ S_0=1, \]

we obtain:

\[ 1= \sum_{\substack{k\geq 0\\ m\geq 0}} \frac{1}{k!m!} \int S_k(x_1,\ldots,x_k)g(x_1)\ldots g(x_k)\,dx_1\ldots dx_k \times \]

\[ \times \int \dot S_m(x_{k+1},\ldots,x_{k+m})g(x_{k+1})\ldots g(x_{k+m})\,dx_{k+1}\ldots dx_{k+m} = \]

\[ = \sum_{(n,k)} \frac{1}{k!(n-k)!} \int S_k(x_1,\ldots,x_k)\dot S_{n-k}(x_{k+1},\ldots,x_n) \times \]

\[ \times g(x_1)\ldots g(x_n)\,dx_1\ldots dx_n . \tag{4.6} \]

Collecting the terms in (4.6) of the same “degree” in \(g(x)\), for \(n=0\) we obtain the identity \((1=1)\), while for \(n>0\) we find:

\[ \sum_k \frac{1}{k!(n-k)!}\int S_k(x_1,\ldots,x_k)\dot S_{n-k}(x_{k+1},\ldots,x_n)\times \]

\[ \times g(x_1)\ldots g(x_n)\,dx_1\ldots dx_n=0 . \tag{4.7} \]

From relation (4.7), for arbitrary \(g(x)\), one still cannot draw the conclusion that the expression

\[ \sum_k \frac{1}{k!(n-k)!}\,S_k(x_1,\ldots,x_k)\dot S_{n-k}(x_{k+1},\ldots,x_n) \tag{4.8} \]

is equal to zero.

Such a conclusion could be made if expression (4.8) proved to be symmetric in all arguments \(x_1,\ldots,x_n\). In reality, however, in each term of (4.8) there is symmetry only within two groups of arguments \(x_1,\ldots,x_k\) and \(x_{k+1},\ldots,x_n\). For the complete symmetrization of an expression of the type (4.8), let us introduce the symbol

\[ P\left(\frac{x_1,\ldots,x_k}{x_{k+1},\ldots,x_n}\right), \]

denoting the sum over all

\[ \frac{n!}{k!(n-k)!} \]

partitions of the set of points \(x_1,\ldots,x_n\) into two subsets of \(k\) and \(n-k\) points. Here permutations within each of these two subsets are not taken into account, since the functions \(S_k\) are symmetric in their arguments.

For example,

\[ P\left(\frac{x_1}{x_2}\right)S_1(x_1)\dot S_1(x_2) = S_1(x_1)\dot S_1(x_2)+S_1(x_2)\dot S_1(x_1), \]

\[ P\left(\frac{x_1,x_2}{x_3}\right)S_2(x_1,x_2)\dot S_1(x_3)= \]

\[ = S_2(x_1,x_2)\dot S_1(x_3)+S_2(x_1,x_3)\dot S_1(x_2)+S_2(x_2,x_3)\dot S_1(x_1). \]

To symmetrize expression (4.7), we rewrite it \(n!\) times, each time changing the notation of the arguments so that the set of points \(x_1,\ldots,x_n\) is permuted in the arguments of the functions \(S_k\) and \(\dot S_{n-k}\) in a new way. Adding the resulting relations, by virtue of the invariance of the weight factor \(g(x_1)\ldots g(x_n)\), we then obtain

\[ \sum_k \int P\left(\frac{x_1,\ldots,x_k}{x_{k+1},\ldots,x_n}\right)S_k(x_1,\ldots,x_k)\times \]

\[ \times \dot S_{n-k}(x_{k+1},\ldots,x_n)g(x_1)\ldots g(x_n)\,dx_1\ldots dx_n=0, \]

whence follows the vanishing of the expression

\[ \sum_k P\left(\frac{x_1,\ldots,x_k}{x_{k+1},\ldots,x_n}\right) S_k(x_1,\ldots,x_k)\dot S_{n-k}(x_{k+1},\ldots,x_n)=0 \]

in view of its symmetry in all \(n\) arguments. Recalling that \(S_0=1\), we obtain from this

\[ \begin{aligned} &S_n(x_1,\ldots,x_n)+\dot S_n(x_1,\ldots,x_n)+{}\\ &\quad+\sum_{(1\leq k\leq n-1)} P\left(\frac{x_1,\ldots,x_k}{x_{k+1},\ldots,x_n}\right) S_k(x_1,\ldots,x_k)\times{}\\ &\quad\quad\times \dot S_{n-k}(x_{k+1},\ldots,x_n)=0. \end{aligned} \tag{4.9} \]

Let us now turn to the causality condition (3.25). We note at the outset that it is more convenient to operate not with the quantity

\[ \frac{\delta S(g)}{\delta g(y)}\dot S(g), \]

which, by virtue of the unitarity condition

\[ S\dot S=1;\qquad \frac{\delta S(g)}{\delta g(y)}\dot S(g)+S(g)\frac{\delta \dot S(g)}{\delta g(y)}=0 \]

is anti-Hermitian, but with the expression

\[ H(y;g)=i\frac{\delta S(g)}{\delta g(y)}\dot S(g), \]

which is, obviously, Hermitian. Using (4.1), we find:

\[ \begin{aligned} H(y;g) &=i\sum_{n_1\geq 0}\frac{1}{n_1!} \int S_{n_1+1}(y,x_1,\ldots,x_{n_1})g(x_1)\ldots g(x_{n_1})\times{}\\ &\quad\times dx_1\ldots dx_{n_1} \sum_{n_2\geq 0}\frac{1}{n_2!} \int \dot S_{n_2}(x_{n_1+1},\ldots,x_{n_1+n_2}) g(x_{n_1+1})\ldots g(x_{n_1+n_2})\times{}\\ &\quad\times dx_{n_1+1}\ldots dx_{n_1+n_2} =\sum_{n>0}\frac{1}{n!}\int H_n(y,x_1,\ldots,x_n)\times{}\\ &\quad\times g(x_1)\ldots g(x_n)\,dx_1\ldots dx_n, \end{aligned} \tag{4.10} \]

where the quantities

\[ \begin{aligned} H_n(y,x_1,\ldots,x_n)={}& i \sum_{0\le k\le n} P\!\left(\frac{x_1,\ldots,x_k}{x_{k+1},\ldots,x_n}\right)\times\\ &\times S_{k+1}(y,x_1,\ldots,x_k)\,\dot S_{n-k}(x_{k+1},\ldots,x_n) =\\ ={}& iS_{n+1}(y,x_1,\ldots,x_n)+ i \sum_{0\le k\le n-1} P\!\left(\frac{x_1,\ldots,x_k}{x_{k+1},\ldots,x_n}\right)\times\\ &\times S_{k+1}(y,x_1,\ldots,x_k)\,\dot S_{n-k}(x_{k+1},\ldots,x_n), \end{aligned} \tag{4.11} \]

are symmetric in all their arguments except the first. Now computing the functional derivative \(\dfrac{\delta}{\delta g(x)}\) of expression (4.9), we obtain:

\[ i\frac{\delta}{\delta g(x)} \left(\frac{\delta S(g)}{\delta g(y)}\,\dot S(g)\right)= \]

\[ =\sum_{n\ge 1}\frac{1}{n!}\int H_n(y,x,x_1,\ldots,x_{n-1})\,g(x_1)\cdots g(x_{n-1})\,dx_1\cdots dx_{n-1}, \]

whence, by virtue of the causality condition (3.25) and on the basis of the symmetry of the functions \(H_n\) in all arguments except the first, it follows that

\[ H_n(y,x_1,x_2,\ldots,x_n)=0, \tag{4.12} \]

if, for at least one \(x_j,\ j=1,\ldots,n\),

\[ y \ge x_j. \]

Thus, with the aid of the covariance, unitarity, and causality conditions to which the matrix \(S(g)\) as a whole is subject, we have derived the corresponding covariance (4.4), unitarity (4.9), and causality (4.12) conditions for the functions \(S_n(x_1,\ldots,x_n)\). We shall now show that the totality of conditions (4.4), (4.9), and (4.12), together with relation (4.3), expressing the correspondence principle, is sufficient to determine the form of the functions

\[ S_n(x_1,\ldots,x_n). \]

We begin with \(S_1(x)\). By the correspondence principle (4.3) we have that

\[ S_1(x)=iL(x). \]

Let us verify the consistency of this relation with the three basic requirements. The fulfillment of the covariance condition (4.4) is obvious,

the causality condition for a single function \(S_1(x)\) cannot yet be formulated, while the unitarity condition (4.9) gives

\[ S_1(x)+\dot S_1(x)=0, \tag{4.13} \]

whence follows the Hermiticity condition

\[ L(x)=\dot L(x) \tag{4.14} \]

for the interaction Lagrangian. Thus, in addition to the locality condition, the Lagrangian \(L(x)\) must also satisfy the Hermiticity condition.

Let us pass to \(S_2(x,y)\).

From the causality condition (4.12), for \(n=1\) we obtain, for \(x>y\),

\[ H_1(x,y)= iS_2(x,y)+ iS_1(x)\dot S_1(y)=0, \]

whence, taking (4.3) into account, it follows that

\[ S_2(x,y)=-S_1(x)\dot S_1(y)=-L(x)L(y) \quad \text{for } x>y. \tag{4.15} \]

Now let \(y>x\). In view of the symmetry of \(S_2\),

\[ S_2(x,y)=S_2(y,x)=-L(y)L(x) \quad \text{for } y>x. \tag{4.16} \]

The domains of definition of relations (4.15) and (4.16) overlap when \(x\sim y\), but here, by virtue of the local character of the operator \(L\), the relation

\[ L(x)L(y)=L(y)L(x) \]

holds, and no contradiction arises between (4.15) and (4.16).

Thus, an expression for the function \(S_2(x,y)\) has been obtained:

\[ S_2(x,y)= \begin{cases} -\,L(x)L(y), & x>y,\\ -\,L(y)L(x), & y>x, \end{cases} \tag{4.17} \]

taking the Hermitian conjugate of which, with (4.14) taken into account, we have

\[ \dot S_2(x,y)= \begin{cases} -\,L(y)L(x), & x>y,\\ -\,L(x)L(y), & y>x. \end{cases} \tag{4.18} \]

The symmetry and polylocality of \(S_2(x,y)\) follow from the form of its writing. It is clear that expression (4.17) satisfies the covariance condition. It is also not difficult to verify that the unitarity condition is fulfilled. In accordance with (4.9), we must check the relation

\[ S_2(x,y)+\dot S_2(x,y)+S_1(x)\dot S_1(y)+S_1(y)\dot S_1(x)=0, \]

the validity of which follows directly from formulas (4.3), (4.13), (4.17), and (4.18). Indeed, we have, for example, for \(x>y\)

\[ S_2(x,y)+\hat S_1(x,y)+S_1(x)\hat S_1(y)+S_1(y)\hat S_1(x)= \]
\[ =-L(x)L(y)-L(y)L(x)+L(x)L(y)+L(y)L(x)=0. \]

One verifies its validity similarly for \(y>x\).

We have thus established that expression (4.17) satisfies all the requirements imposed on \(S_2\).

It is now expedient to introduce the concept of an ordered, or chronological, product of operators. For the time being we shall introduce it only for local operators.

The chronological product of a number of local operators \(\Phi_1(x_1),\ldots,\Phi_n(x_n)\) is denoted by the symbol \(T[\Phi_1(x_1)\ldots\Phi_n(x_n)]\) and, by definition, is equal to the ordinary product of these operators taken in a definite order, corresponding to the decrease of the time components of the arguments of the factors from left to right, i.e.

\[ T[\Phi_1(x_1)\ldots\Phi_n(x_n)] =\Phi_{j_1}(x_{j_1})\Phi_{j_2}(x_{j_2})\ldots\Phi_{j_n}(x_{j_n}), \tag{4.19} \]

where the sequence \(x_{j_1},\ldots,x_{j_n}\) is the sequence of arguments \(x_1,\ldots,x_n\) arranged in the order of decreasing time components, i.e.

\[ x^0_{j_1}>x^0_{j_2}>\cdots>x^0_{j_n}. \]

For brevity, we shall often call the chronological product a \(T\)-product. We shall also need the concept of the antichronological product, by definition corresponding to an increase of the time components of the arguments from left to right. Taking, for example, the Hermitian conjugate of the right-hand side of (4.19), we obtain the expression

\[ \hat\Phi_{j_n}(x_{j_n})\ldots \hat\Phi_{j_1}(x_{j_1}), \]

\[ x^0_{j_n}\leq x^0_{j_{n-1}}\leq\cdots\leq x^0_{j_1}, \]

which it is convenient to call the antichronological product (or \(T^*\)-product) of the operators \(\hat\Phi_1(x_1)\ldots\hat\Phi_n(x_n)\). We therefore put

\[ T^*(\Phi_1(x_1)\ldots\Phi_n(x_n)) =\Phi_{j_1}(x_{j_1})\Phi_{j_2}(x_{j_2})\ldots\Phi_{j_n}(x_{j_n}), \tag{4.20} \]

\[ x^0_{j_1}\leq x^0_{j_2}\leq\cdots\leq x^0_{j_n}. \]

We shall show that the definition of the \(T\)-product is covariant, despite the distinguished role of time.

For the value of the \(T\)-product the chronological order of the points \(x_1,\ldots,x_n\) is essential; this order may change under passage from one Lorentz frame of reference to another under an infinitesimal rotation. It is not difficult, however, to verify that such a transition does not change the value of the \(T\)-product. To see this, let us imagine that we carry out the transformation of interest to us by means of a large number of small transformations, which is always possible in view of the continuous character of the transformations of the extended Lorentz group. The process of changing the chronological order of the points \(x_1,\ldots,x_n\) will then be split into a certain number of stages, at each of which the time order within some group \(x'_j,\ldots,x'_k\) (of two or more) points will change simultaneously. But the chronological order of several points \(x_j,\ldots,x_k\) can be changed by a Lorentz rotation only in the case of mutual spatial likeness of these points. The corresponding operators \(\varphi_j(x_j),\ldots,\varphi_k(x_k)\), by virtue of the locality property, in this case commute with one another, and their order is immaterial. Therefore at each stage of the transformation the value of the \(T\)-product does not change, and consequently it does not change at all.

It is also not difficult to note that, according to definition (4.19), local operators may be commuted under the sign of the \(T\)-product without changing its value. Hence it follows, in particular, that if all the operators \(\varphi\) are identical, then the \(T\)-product turns out to be a symmetric function of its arguments.

Returning to formula (4.17), we see that with the aid of the \(T\)-product it may be written in the form

\[ S_s(x,y)=-T[L(x),L(y)]. \tag{4.21} \]

Correspondingly, (4.18) assumes the form

\[ \dot S_s(x,y)=-T^*[L(x),L(y)]. \tag{4.22} \]

We now show that, in general, the expression

\[ S_n(x_1,\ldots,x_n)=i^n T[L(x_1)\ldots L(x_n)], \tag{4.23} \]

which is the natural generalization of formulas (4.3) and (4.21), satisfies all the formal conditions imposed on \(S_n\). Fulfilment of the conditions of symmetry, covariance, and polylocality is now evident and requires verification only of the conditions of unitarity and causality.

It will be more convenient for us, however, to operate for this purpose not with formula (4.23), but with the operator \(S(g)\) as a whole. Substituting (4.23) into (4.1), we obtain for \(S(g)\)

\[ S(g)=1+\sum_{n>1}\frac{i^n}{n!}\int T[L(x_1),\ldots,L(x_n)]\times \]

\[ \times g(x_1)\ldots g(x_n)\,dx_1\ldots dx_n. \tag{4.24} \]

This expression represents the expansion of the operator \(S(g)\) in a functional series in powers of \(g(x)\). At present we are interested in certain properties of the coefficients of the expansion. Just as in the theory of special functions it is often more convenient to derive mutual properties of functions not from their specific structure, but from some common generating function, in the present situation it proves simpler to verify the unitarity and causality conditions for the whole series (4.24) as a whole.

For this purpose let us write expression (4.24) in a somewhat different form. We represent the \(n\)-th term of the series in the following form:

\[ \frac{i^{n}}{n!}\,T\left[\int L(x_1)g(x_1)\,dx_1 \ldots \int L(x_n)g(x_n)\,dx_n\right] = \]

\[ =\frac{i^{n}}{n!}\,T\left[\left(\int L(x)g(x)\,dx\right)^n\right]. \]

The series (4.24) can now be formally summed by introducing the \(T\)-exponential\({}^{17}\)

\[ S(g)=T\left[1+\sum_{n\geqslant 1}\frac{i^n}{n!}\left(\int L(x)g(x)\,dx\right)^n\right] \equiv T\left(e^{\,i\int L(x)g(x)\,dx}\right). \tag{4.25} \]

We thus obtain a new expression for the scattering matrix.

The important concept of the \(T\)-exponential can also be approached from another side. Let us divide the region in which the interaction, described by the function \(g(x)\), is switched on into an infinitely large number of infinitely thin layers \(\Delta_j\) by spacelike surfaces \(t=\mathrm{const}\). We then have

\[ T\left(e^{\,i\int L(x)g(x)\,dx}\right) = T\left(e^{\,i\sum_j \int_{\Delta_j} L(x)g(x)\,dx}\right) = \]

\[ = T\left(\prod_j e^{\,i\int_{\Delta_j} L(x)g(x)\,dx}\right). \]

It is therefore natural to define the \(T\)-exponential (4.25) as the limit of \(T\)-products

\[ T\left(e^{\,i\int L(x)g(x)\,dx}\right) = \lim_{\Delta_j\to 0} T\left[\prod_j\left(1+i\int_{\Delta_j} L(x)g(x)\,dx\right)\right]. \tag{4.26} \]

With the aid of representation (4.26), the unitarity property of the matrix \(S(g)\) becomes evident.

In fact, the \(T\)-product (4.26) is the ordinary product

\[ \prod_j \left(1+i\int_{\Delta_j} L(x)g(x)\,dx\right), \]

taken in the proper chronological order of the layers \(\Delta_j\). But each factor of this product, for sufficiently small \(\Delta_j\), is unitary up to quantities of higher order of smallness. Therefore the entire product is unitary. Thus the unitarity of the limit of this product, equal to \(S(g)\), has been proved.

Let us now turn to the verification of the causality condition. Computing the variational derivative of \(S(g)\) at the point \(y\), we find:

\[ -i\,\frac{\delta S(g)}{\delta g(y)} = T\left(L(y)e^{\,i\int L(z)g(z)\,dz}\right). \]

We divide four-dimensional space into two parts \(G_+\) and \(G_-\) by the space-like surface

\[ x^0=\mathrm{const}=y^0. \]

Let \(G_+\) lie “in the future,” and \(G_-\) “in the past.” We then have

\[ -i\,\frac{\delta S(g)}{\delta g(y)} = T\left( L(y)e^{\,i\int_{G_+} L(z)g(z)\,dz+i\int_{G_-}L(z)g(z)\,dz} \right) = \]

\[ = T\left( L(y)e^{\,i\int_{G_+} L(z)g(z)\,dz} \right) T\left( e^{\,i\int_{G_-} L(z)g(z)\,dz} \right), \tag{4.27} \]

On the other hand, we obtain in a completely analogous way:

\[ S(g)= T\left( e^{\,i\int_{G_+} L(z)g(z)\,dz+i\int_{G_-} L(z)g(z)\,dz} \right) = \]

\[ = T\left( e^{\,i\int_{G_+} L(z)g(z)\,dz} \right) \cdot T\left( e^{\,i\int_{G_-} L(z)g(z)\,dz} \right), \]

and therefore

\[ S^{*}(g)= T^{*}\left( e^{\,i\int_{G_-} L(z)g(z)\,dz} \right) \cdot T^{*}\left( e^{\,i\int_{G_+} L(z)g(z)\,dz} \right). \]

Taking into account the unitarity property of the expression

\[ T\left( e^{\,i\int_{G_-} L(z)g(z)\,dz} \right); \]

whence, on the basis of (4.27), we find:

\[ -i\,\frac{\delta S(g)}{\delta g(y)}\,\overset{*}{S}(g) = T\left(L(y)e^{\,i\int_{G_+} L(z)g(z)\,dz}\right) T\left(e^{\,i\int_{G_-} L(z)g(z)\,dz}\right). \tag{4.28} \]

Thus,

\[ -i\,\frac{\delta S(g)}{\delta g(y)}\,\overset{*}{S}(g) \]

does not depend on the behavior of the function \(g(x)\) in the region \(G_-\), i.e. for \(x^0<y^0\).

By considerations of covariance, this relation also holds in the case of a space-like separation of the points \(x\) and \(y\) (for \(x\sim y\)). The verification of the causality condition is thereby completed.

The proofs given above of the causality and unitarity of the operator are very simple and transparent. It is necessary, however, to note that from a purely mathematical point of view they are not fully consistent. Indeed, in the course of the argument we related the question of whether the elementary relations (4.9) and (4.12) hold for the product (4.23) to the entirely obscure questions of summing the series (4.1) as a whole, of passage to the limit, and so on. Strictly speaking, all these elements are not at all required for the proof.

The point is that, instead of operating with the \(T\)-exponential, we may introduce the “\(T\)-exponential accurate up to a prescribed degree \(g\),” and then all questions concerning summation of the series are automatically removed.

We have thus convinced ourselves that the expression

\[ S_n(x_1,\ldots,x_n)=i^n T\bigl(L(x_1)\cdots L(x_n)\bigr) \]

is admissible in the sense of satisfying all the conditions imposed on \(S_n\). It turns out, however, that this expression is not the most general expression satisfying all the imposed conditions. Let us therefore consider the question of constructing the most general expression for \(S_n(x_1,\ldots,x_n)\) satisfying the conditions of symmetry, covariance, causality, and unitarity, and thereby completely solve the posed problem of constructing the operator \(S(g)\).

For this purpose we first examine the procedure for determining the function \(S_n(x_1,\ldots,x_n)\) from the previously given functions \(S_1,S_2,\ldots,S_{n-1}\). By the unitarity condition (4.9), \(S_n\) is determined through them up to a certain anti-Hermitian operator, which we shall denote by \(i\Lambda_n(x_1,\ldots,x_n)\). In accordance with the symmetry condition, the quantity \(\Lambda_n(x_1,\ldots,x_n)\) must be a symmetric function of its arguments \(x_1,\ldots,x_n\). By the condition

because of causality (4.12) the operator function \(S_n(x_1,\ldots,x_n)\) is completely determined by the preceding functions in the domain of definition of its arguments in which

\[ x_1 > \text{ at least one of the } x_j,\quad j=2,3,\ldots,n . \]

Therefore, in the indicated domain, the anti-Hermitian operator \(iA_n(x_1,\ldots,x_n)\) must be equal to zero. From the property of its symmetry in all arguments it follows that it is also equal to zero if, for at least one pair of arguments \(x_i\) and \(x_j\),

\[ x_i \ne x_j \]

and can, consequently, differ from zero only when all the arguments coincide completely,

\[ x_1=x_2=\cdots=x_n . \]

Thus, from the conditions of causality, unitarity, and symmetry it follows that the Hermitian operator \(A_n(x_1,\ldots,x_n)\) is a quasilocal operator, in the sense of the definition given in § 2, and its coefficient functions have the form

\[ Z\left(\ldots\frac{\partial}{\partial x_i}\ldots\right)\delta(x_1-x_2)\cdots\delta(x_1-x_n), \]

where, by considerations of translational invariance, \(Z\) cannot depend on \(x_1\).

Thus, it has been established that the conditions of invariance, symmetry, unitarity, and causality, for given \(S_1,S_2,\ldots,S_{n-1}\), determine \(S_n\) up to \(iA_n\), where \(A_n(x_1,\ldots,x_n)\) is a Hermitian symmetric quasilocal operator transforming as a scalar. Therefore, in order to obtain an expression for \(S_1,S_2,\ldots,S_n\), in addition to the local operator \(L(x)\) it is necessary also to specify a chain of quasilocal operators

\[ A_2(x_1,x_2),\ldots,A_n(x_1,\ldots,x_n). \tag{4.29} \]

We have arrived at results that, at first glance, are somewhat strange. For the complete determination of the matrix \(S(g)\), specifying the Lagrangian of the interaction turns out to be insufficient, and it is necessary to specify also an infinite chain of quasilocal operators (4.29).

To clarify the essence of the situation, let us approach the question under consideration from a somewhat different side. For this purpose consider the expression

\[ T\left(e^{i\int L(x;g)g(x)\,dx}\right), \tag{4.30} \]

where the “Lagrangian” \(L(x;g)\) is determined by the relation

\[ L(x;g)=L(x)+ \sum_{\nu\geq 2}\frac{1}{\nu!}\int \Lambda_\nu(x,x_1,\ldots,x_{\nu-1}) g(x_1)\cdots g(x_{\nu-1})\,dx_1\cdots dx_{\nu-1}. \tag{4.31} \]

By virtue of the quasilocal character of the functions \(\Lambda_\nu\), all the integrations in (4.31) are removed, and \(L(x;g)\) in fact depends on the field functions \(u(x)\) at the point \(x\). Therefore \(L(x;g)\) is a local operator which, besides the field operators \(u(x)\), depends on the function \(g(x)\), which may be regarded as a “classical” field. Consequently, expression (4.30) satisfies all the conditions imposed on \(S(g)\), including the correspondence condition (4.3), and may be regarded as the scattering matrix \(S(g)\). Expanding (4.30) in a series in powers of \(g\), we therefore obtain certain expressions for \(S_n(x_1,\ldots,x_n)\) satisfying all the imposed conditions. We have:

\[ T\left(e^{\,i\int L(x;g)g(x)\,dx}\right) = 1+\sum_{m\geq 1}\frac{i^m}{m!}\int T\{L(x_1;g)\cdots L(x_m;g)\} \times \]

\[ \times g(x_1)\cdots g(x_m)\,dx_1\cdots dx_m. \]

Substituting here the expansions (4.31), we obtain:

\[ T\left(e^{\,i\int L(x;g)g(x)\,dx}\right) = 1+\sum_{\substack{m\geq 1\\(\nu_1\geq 0,\ldots,\nu_m\geq 0)}} \frac{i^m}{m!}\, \frac{1}{\nu_1!\cdots\nu_m!}\times \]

\[ \times \int T\{\Lambda_{\nu_1}(x_1,\ldots,x_{\nu_1})\cdots \Lambda_{\nu_m}(x_{\nu_1+\cdots+\nu_{m-1}+1},\ldots \]

\[ \ldots,x_{\nu_1+\cdots+\nu_m})\} \,g(x_1)\cdots g(x_{\nu_1+\cdots+\nu_m})\times \]

\[ \times dx_1\cdots dx_{\nu_1+\cdots+\nu_m}, \]

where, for symmetry, we have put \(L(x)=\Lambda_1(x)\). The symbol

\[ T\{\Lambda_{\nu_1}(x_1,\ldots,x_{\nu_1})\cdots \Lambda_{\nu_s}(x_i,\ldots,x_{i+\nu_s})\} \]

denotes the product of the operators \(\Lambda_{\nu_1},\ldots,\Lambda_{\nu_s}\), taken in chronological order of the time arguments. The multiplicity of the arguments of each \(\Lambda_\nu\) should not trouble us, since by definition \(\Lambda_\nu\) is nonzero only when its arguments coincide.

Let us now rearrange this series according to powers of \(g(x)\), separating out the terms in which \(g(x)\) enters to a definite \(n\)-th degree and which

contain exactly \(n\) integrations:

\[ T\left(e^{\,l\int L(x;g)g(x)\,dx}\right) =1+\sum_{n\geq 1}\ \sum_{\substack{1\leq m\leq n\\ \sum \nu_i=n}} \frac{l^m}{m!}\,\frac{1}{\nu_1!\cdots \nu_m!}\times \]

\[ {}\times \int T\{\Lambda_{\nu_1}(x_1,\ldots,x_{\nu_1})\cdots \Lambda_{\nu_m}(x_{\nu_1}+\cdots+\nu_{m-1}+1\ldots x_n)\}\times \]

\[ {}\times g(x_1)\cdots g(x_n)\,dx_1\cdots dx_n. \]

The latter expansion differs further from expansion (4.1) by the nonsymmetric character of the coefficients at the powers of the function \(g(x)\). To symmetrize them, using the symmetry of the weight factor \(g(x_1)\cdots g(x_n)\) for each given \(n\) and the symmetry of the functions \(\Lambda_\nu\), we perform \(\dfrac{n!}{\nu_1!\cdots\nu_m!}\) substitutions of the designations of the variables \(x_1,\ldots,x_n\), so that the sum of all the expressions obtained, taking into account the symmetries of \(\Lambda_\nu\), would be a symmetric function of all arguments \(x_1,\ldots,x_n\). Dividing the result by the number \(\dfrac{n!}{\nu_1!\cdots\nu_m!}\), we arrive at the expression

\[ T\left(e^{\,l\int L(x;g)g(x)\,dx}\right) =1+\sum_{n\geq 1}\ \sum_{\substack{1\leq m\leq n\\ \sum \nu_i=n}} \frac{l^m}{n!m!}\times \]

\[ {}\times \int P(x_1,\ldots,x_{\nu_1}\mid x_{\nu_1}+1\cdots\mid\cdots x_n)\times \]

\[ {}\times T\{\Lambda_{\nu_1}(x_1,\ldots,x_{\nu_1})\cdots \Lambda_{\nu_m}(\ldots x_n)\}\,g(x_1)\cdots g(x_n)\,dx_1\cdots dx_n, \]

where \(P(x_1,\ldots,x_{\nu_1}\mid x_{\nu_1}+1\cdots\mid\cdots x_n)\) is the symmetrization operator over arbitrary partitions of the set of \(n\) points into \(\dfrac{n!}{\nu_1!\cdots\nu_m!}\) \((\sum \nu_i=n)\) all possible partitions into \(\nu_1,\nu_2,\ldots,\nu_m\) points. This operator is a natural generalization of the operator

\[ P\left(\frac{x_1\cdots x_\nu}{x_{\nu+1}\cdots x_n}\right) = P(x_1,\ldots,x_\nu\mid x_{\nu+1}\cdots x_n). \]

Thus, the matrix (4.29) is represented in the form

\[ T\left(e^{\,l\int L(x;g)g(x)\,dx}\right)= \]

\[ =1+\sum_{n\geq 1}\frac{1}{n!}\int S_n(x_1,\ldots,x_n)g(x_1)\cdots g(x_n)\,dx_1\cdots dx_n, \]

where the coefficients \(S_n\) have the form

\[ S_n(x_1,\ldots,x_n)= \sum_{\substack{1\leq m\leq n\\ \sum \nu_i=n}} \frac{l^m}{m!}\, P(x_1,\ldots,x_{\nu_1}\mid\cdots\mid\cdots x_n)\times \]

\[ {}\times T\{\Lambda_{\nu_1}(x_1,\ldots,x_{\nu_1})\cdots \Lambda_{\nu_m}(\ldots x_n)\}. \]

Recalling that \(\Lambda_1(x)=L(x)\), we can rewrite \(S_n\) in the form

\[ \begin{aligned} S_n(x_1,\ldots,x_n)= i^n T\,[L(x_1)\ldots L(x_n)] &+ \\ + \sum_{\substack{(2\le m\le n-1)\\ \nu_1+\cdots+\nu_m=n}} \frac{i^m}{m!}\,P(x_1,\ldots,x_{\nu_1}|\ldots|\ldots x_n)\;&\times \\ \times T\,[\Lambda_{\nu_1}(x_1,\ldots,x_{\nu_1})\ldots \Lambda_{\nu_m}(1\ldots x_n)] &+ i\Lambda_n(x_1,\ldots,x_n). \end{aligned} \tag{4.32} \]

Let us consider several simple particular cases of the formula obtained (4.32).

For \(n=2\) we obtain:

\[ S_2(x_1,x_2)=i^2T\,[L(x_1)L(x_2)]+i\Lambda_2(x_1,x_2). \tag{4.33a} \]

For \(n=3\), respectively, we have:

\[ \begin{aligned} S_3(x_1,x_2,x_3)= i^3T\,[L(x_1)L(x_2)L(x_3)] &+ \\ + \sum_{\substack{(\nu_1+\nu_2=3)\\ m=2}} \frac{i^2}{2!}\,P(x_1,x_{\nu_1}|\ldots x_3)\, T\,[\Lambda_{\nu_1}(x_1,x_{\nu_1})\Lambda_{\nu_2}(\ldots x_3)] &+ \\ +\Lambda_3(x_1,x_2,x_3) = -iT\,[L(x_1)L(x_2)L(x_3)] &- \\ - T\,[L(x_1)\lambda_2(x_2,x_3)] - T\,[L(x_2)L(x_1,x_3)] &- \\ - T\,[L(x_3)L(x_1,x_2)] +i\Lambda_3(x_1,x_2,x_3). \end{aligned} \tag{4.33b} \]

We see from this that each subsequent function \(S_n\) is expressed in terms of the preceding \(S_1,\ldots,S_{n-1}\), up to a symmetric, anti-Hermitian, quasilocal operator \(i\Lambda_n(x_1,\ldots,x_n)\). Therefore expression (4.32) is the most general expression for \(S_n\), while the expression

\[ S(g)=T\left(e^{\,i\int L(x;g)\,g(x)\,dx}\right), \tag{4.34} \]

where \(L(x;g)\) is determined by the expansion (4.31), is the most general expression for the scattering matrix.

Thus, the chain of quasilocal operators (4.29), which must be specified for the complete determination of the scattering matrix \(S(g)\), can in fact be included in the “Lagrangian” of the interaction.

The question now arises of the physical meaning of the linear combination (4.31) \(L(x)\) and of the integrals of the quasilocal operators, playing the role of the most complete admissible Lagrangian.

The point is that in our reasoning a nonphysical operation of “switching on” the interaction was used. It then turned out that part of the Lagrangian was “switched on” in the first approximation, part in the second, part in the \(n\)-th, and the interaction Lagrangian was split into a chain of separate pieces. Physical meaning, however, belongs to the situation when the interact-

... is included completely. In this case the function \(\xi(x)\) is equal to unity, and expressions (4.31) and (4.34) take the form

\[ L(x;1)=L(x)+ +\sum_{\nu\geq 2}\frac{1}{\nu!}\int \Lambda_\nu(x,x_1,\ldots,x_{\nu-1})\,dx_1\ldots dx_{\nu-1}, \tag{4.35} \]

\[ S(1)=T\left(e^{\,i\int L(x;1)\,dx}\right). \tag{4.36} \]

Therefore the real scattering matrix \(S(1)\) is completely characterized by the real Lagrangian of the interaction of the system \(L(x;1)\), which in perturbation theory is sometimes represented in the form of a series of separate terms.

In the usual exposition of field theory, starting from the Schrödinger equation, one obtains a representation for the scattering matrix in the form

\[ T\left(e^{-i\int H(x)\,dx}\right), \tag{4.37} \]

where \(H(x)\) is the density of the interaction Hamiltonian.

\(H(x)\) coincides with \(-L(x)\) only in especially simple cases, when \(L(x)\) does not depend on derivatives of the field functions. In the general case \(H(x)\), in addition to the term \(-L(x)\), also contains certain noncovariant terms. However, as a result of a rather complicated procedure\({}^{18}\) these terms are completely eliminated, and expression (4.37) is reduced to the form (4.36).

§ 5. EXPANSION OF CHRONOLOGICAL PRODUCTS

Having an explicit expression for the scattering matrix, we can proceed to the computation of its matrix elements for various states.

In the course of this computation we shall have to reduce the terms of the matrix to normal form, i.e. to such a form in which, in the individual summands, all annihilation operators stand on the right and the creation operators on the left.

We shall therefore concern ourselves with the expression of the \(T\)-product of local operators \(L(x)\) in terms of normal products of the corresponding field operators. For this purpose it will be convenient for us to extend the concept of the \(T\)-product to the case of a general system of linear operators, defined in § 2.

Let us define the chronological, or ordered, product

\[ T\bigl(A_1(x_1)\ldots A_n(x_n)\bigr) \]

of the linear operators \(A_1(x_1)\ldots A_n(x_n)\) as the ordinary product in chronological order, multiplied by \(\varepsilon=(-1)^\eta\), where \(\eta\)—

parity of Fermi permutations in passing from the order \(1,\ldots,n\) to chronological order, i.e.,
\[ T\bigl(A_1(x_1)\ldots A_n(x_n)\bigr)=\varepsilon A_{j_1}(x_{j_1})\ldots A_{j_n}(x_{j_n}) \tag{5.1} \]
for
\[ x^0_{j_1}>x^0_{j_2}>\ldots>x^0_{j_n}; \qquad \varepsilon=(-1)^\eta, \]
where \(\eta\) is the parity of the permutation of Fermi operators in passing from the order \((1,2,\ldots,n)\) to the order \((j_1,j_2,\ldots,j_n)\).

The recipe for expanding such products is given by “Wick’s theorem for \(T\)-products,” which is an analogue of “Wick’s theorem for ordinary products.” Before proceeding to the proof of this theorem, let us introduce the important notion of “chronological pairing of operators.”

To this end, consider (5.1) in the case \(n=2\). Then
\[ T\bigl(A_1(x_1)A_2(x_2)\bigr)= \begin{cases} A_1(x_1)A_2(x_2), & x^0_1>x^0_2,\\ \varepsilon A_2(x_2)A_1(x_1), & x^0_2>x^0_1. \end{cases} \]

This expression, in accordance with the definition of ordinary pairing,
\[ A_1(x_1)A_2(x_2)=:A_1(x_1)A_2(x_2):+\underbrace{A_1(x_1)A_2(x_2)} \]
can be transformed into the form
\[ \begin{aligned} T\bigl(A_1(x_1)A_2(x_2)\bigr) &= \begin{cases} :A_1(x_1)A_2(x_2):+\underbrace{A_1(x_1)A_2(x_2)}, & x^0_1>x^0_2,\\ \varepsilon :A_2(x_2)A_1(x_1):+\varepsilon \underbrace{A_2(x_2)A_1(x_1)}, & x^0_2>x^0_1 \end{cases} \\ &= :A_1(x_1)A_2(x_2):+\varepsilon \underbrace{A_2(x_2)A_1(x_1)}, \quad x^0_2>x^0_1 . \end{aligned} \tag{5.2} \]

We see from this that in every case \(T\bigl(A_1(x_1)A_2(x_2)\bigr)\) differs from \(:A_1(x_1)A_2(x_2):\) by a \(c\)-number, which we shall call the chronological pairing and denote by the symbol \(\overline{A_1(x_1)A_2(x_2)}\), i.e., by definition
\[ T\bigl(A_1(x_1)A_2(x_2)\bigr)=:A_1(x_1)A_2(x_2):+\overline{A_1(x_1)A_2(x_2)}, \tag{5.3} \]
where, according to (5.2),
\[ \overline{A_1(x_1)A_2(x_2)}= \begin{cases} \underbrace{A_1(x_1)A_2(x_2)}, & x^0_1>x^0_2,\\ \varepsilon \underbrace{A_2(x_2)A_1(x_1)}, & x^0_2>x^0_1. \end{cases} \tag{5.4} \]

Let us first of all note the characteristic difference of chronological pairing. Under the sign of chronological pairing one may [[unclear: word beginning with “iz-”]]-

change the order of the factors, as well as under the sign of the normal product, i.e.

\[ \overline{A_1(x_1)A_2(x_2)}=\varepsilon \overline{A_2(x_2)A_1(x_1)}, \]

which follows directly from (5.4).

Let us now define chronological contractions for the principal types of fields we have investigated.

For the complex scalar field, using the expressions for the ordinary contractions (of type (2.8)), we find:

\[ \overline{\varphi(x)\varphi^*(y)}= \begin{cases} -iD^{(-)}(x-y), & x^0>y^0,\\ -iD^{(-)}(y-x)=iD^{(+)}(x-y), & y^0>x^0. \end{cases} \tag{5.5} \]

Thus we see that, for \(x^0>y^0\) and \(y^0>x^0\), and also for \(x\simeq y\), where both forms coincide, the chronological contraction

\[ \overline{\varphi(x)\varphi^*(y)} \]

coincides with the causal Green function \(D^c(x-y)\) (up to the factor \(i^{-1}\)). We shall therefore agree to consider that this coincidence also holds in an infinitely small neighborhood of the point \(x=y\), i.e. always

\[ \overline{\varphi(x)\varphi^*(y)} =\frac{1}{i}D^c(x-y) =\frac{1}{(2\pi)^4 i}\int \frac{e^{ik(x-y)}\,dk}{m^2-k^2-i\varepsilon}. \tag{5.6} \]

As is evident, in introducing contractions there is a certain arbitrariness. Indeed, the relations (5.5) determine it only for \(x\ne y\).

The rules for integrating the given expression in an infinitely small neighborhood of the point \(x=y\) may be fixed arbitrarily. One can always, for example, add to the right-hand side of (5.6) any coefficient function of the quasi-local operator

\[ P\!\left(\frac{\partial}{\partial x}\right)\delta(x-y), \]

where \(P\!\left(\frac{\partial}{\partial x}\right)\) is any polynomial in \(\frac{\partial}{\partial x^\alpha}\). This necessity of an additional definition of the contraction in an infinitely small neighborhood of the point \(x=y\) is a partial manifestation of the arbitrariness contained in the \(T\)-product. The point is that the \(T\)-product itself is specified by our formal “definition” (4.19) only for noncoincident values of its arguments. It is therefore necessary, in general, to additionally define the \(T\)-products in the corresponding infinitely small neighborhoods of the points of coincidence of the arguments, by specifying the rules for integrating their coefficient functions, i.e., in other words, one must define the coefficient functions of the \(T\)-products as integrable improper functions.

Thus, we arrive at the conclusion that it is necessary not only to choose the interaction Lagrangian, but also simultaneously to specify the \(T\)-products.

It should be noted, however, that the influence of changing the \(T\)-products on \(S(g)\) can be taken into account by changing the Lagrangian \(L(x)\). Indeed, when the \(T\)-products of field functions are changed, we thereby introduce into the \(T\)-products of the interaction Lagrangians various quasilocal operators, which, as was shown in the preceding paragraph, reduces to adding certain expressions to the interaction Lagrangian.

Thus we arrive at the conclusion that, in order to obtain matrix elements of the scattering matrix \(S(g)\) that determine the structure of physical processes, it is necessary to specify simultaneously the interaction Lagrangian and the rules for integrating \(T\)-products. If the rules for integrating \(T\)-products have already been fixed, then the interaction Lagrangian \(L(x)\) must be chosen with respect to these rules.

The dependence of the form of the Lagrangian on certain additional considerations is by no means a specific feature of secondary quantization. In classical physics, for example, in order to fix the form of the Lagrangian one must first choose independent dynamical variables (compare, for example, the treatment of the scalar field in the usual way and in Kemmer’s formalism \(^{19}\)).

Thus, we must first additionally define all contractions, as well as their products, so that the latter turn out to be integrable functions. Then the \(T\)-products will be completely specified, and it will become possible to fix the Lagrangian.

Questions connected with the problem of regularization of the \(S\)-matrix will be considered in detail in the following article. Here we shall restrict ourselves to additionally defining contractions of field functions. We shall agree to assume that if, for \(x \ne y\), a contraction coincides with a certain Green’s function

\[ \Delta_{\alpha}^{c}(x-y)=P_{\alpha}\!\left(\frac{\partial}{\partial x}\right)D^{c}(x-y), \]

then it coincides with this function also in an infinitely small neighborhood of the point \(x=y\).

Thus, carrying out the corresponding calculations for the electromagnetic and spinor fields, we find that the chronological contractions of the field functions of these fields are

\[ \overline{A_m(x)A_n(y)}= \begin{cases} i g^{mn}D_0^{(-)}(x-y), & x^0>y^0,\\ - i g^{mn}D_0^{(+)}(x-y), & y^0>x^0, \end{cases} \]

\[ \overline{\psi(x)\bar{\psi}(y)}= \begin{cases} - iS^{(-)}(x-y), & x^0>y^0,\\ + iS^{(+)}(x-y), & y^0>x^0. \end{cases} \]

for \(x \ne y\) coincide with the causal functions of the corresponding fields. We shall therefore assume that always

\[ \overline{A_m(x)A_n(y)} = i g_{mn}D_0^c(x-y) = \frac{g_{mn}}{(2\pi)^4 i}\int \frac{dk\, e^{ik(x-y)}}{k^2+i\varepsilon}, \tag{5.7} \]

\[ \overline{\psi(x)\bar\psi(y)} = \frac{1}{i}S^c(x-y) = \frac{1}{(2\pi)^4 i}\int \frac{dp\, e^{ip(x-y)}(\hat p-m)}{p^2-m^2+i\varepsilon}. \tag{5.8} \]

As was already noted in § 1, the functions of the fields usually considered satisfy the commutation relations

\[ [u_\alpha(x),u_\beta^{+}(y)] = \frac{1}{i}\Delta_{\alpha\beta}(x-y) = P_{\alpha\beta}\!\left(\frac{\partial}{\partial x}\right)D(x-y), \]

\[ [u_\alpha^{(-)}(x),u_\beta^{(+)+}(y)] = \frac{1}{i}\Delta_{\alpha\beta}^{(-)}(x-y) = P_{\alpha\beta}\!\left(\frac{\partial}{\partial x}\right)D^{(-)}(x-y). \]

Therefore, computing the ordinary contraction, we obtain:

\[ \overline{u_\alpha(x)u_\beta^{+}(y)} = \frac{1}{i}\Delta_{\alpha\beta}^{(-)}(x-y), \]

whence, in the standard way, we find for the chronological contraction the expression

\[ \overline{u_\alpha(x)u_\beta^{+}(y)} = \begin{cases} -i\Delta_{\alpha\beta}^{(-)}(x-y), & x^0>y^0,\\[2mm] +i\Delta_{\alpha\beta}^{(+)}(x-y), & y^0>x^0, \end{cases} \]

which for \(x\ne y\) coincides with

\[ \frac{1}{i}\Delta_{\alpha\beta}^{c}(x-y) = \frac{1}{i}P_{\alpha\beta}\!\left(\frac{\partial}{\partial x}\right)D^c(x-y). \]

We shall therefore put, for any \(x\) and \(y\),

\[ \overline{u_\alpha(x)u_\beta^{+}(y)} = \frac{1}{i}\Delta_{\alpha\beta}^{c}(x-y). \tag{5.9} \]

In a number of cases derivatives of field functions may enter into the interaction Lagrangians. It is therefore expedient to introduce a condition for the complete determination of their contractions as well.

Noting that for \(x^0<y^0\)

\[ \overline{ \frac{\partial^{k}u_\alpha(x)} {(\partial x^0)^{k_0}\cdots(\partial x^3)^{k_3}} \cdot \frac{\partial^{l}u_\beta^{+}(y)} {(\partial y^0)^{l_0}\cdots(\partial y^3)^{l_4}} } = \]

\[ = \frac{\partial^{k}} {(\partial x^0)^{k_0}\cdots(\partial x^3)^{k_3}} \cdot \frac{\partial^{l}} {(\partial y^0)^{l_0}\cdots(\partial y^3)^{l_4}} \,\overline{u_\alpha(x)u_\beta^{+}(y)}. \]

we shall put, by definition,

\[ \overline{ \frac{\partial^{k} u_{\alpha}(x)} {(\partial x^{0})^{k_{0}}\ldots(\partial x^{3})^{k_{3}}} \cdot \frac{\partial^{q} u_{\beta}^{+}(y)} {(\partial y^{0})^{q_{0}}\ldots(\partial y^{3})^{q_{3}}} } = \frac{1}{i}\, \frac{\partial^{k}} {(\partial x^{0})^{k_{0}}\ldots(\partial x^{3})^{k_{3}}} \cdot \frac{\partial^{q}} {(\partial x^{0})^{q_{0}}\ldots(\partial y^{3})^{q_{3}}} \,\Delta^{c}_{\alpha\beta}(x-y). \tag{5.10} \]

Having defined chronological contraction, we pass to the formulation of Wick’s theorem for \(T\)-products.

“Wick’s theorem for \(T\)-products” consists in the assertion that the \(T\)-product of a system of \(n\) linear operators is equal to the sum of their normal products with all possible chronological contractions (including the term without contractions).

The proof reduces essentially to the proof of Wick’s theorem for ordinary products. Indeed, according to definition (5.1), the \(T\)-product is equal to some ordinary product

\[ T\bigl(A_{1}(x_{1})\ldots A_{n}(x_{n})\bigr) = \varepsilon A_{j_{1}}(x_{j_{1}})\ldots A_{j_{n}}(x_{j_{n}}). \]

Applying Wick’s theorem to this ordinary product, we see that it is equal to the sum of normal products of the operators
\(A_{j_{1}}(x_{j_{1}})\ldots A_{j_{n}}(x_{j_{n}})\) with all possible ordinary contractions. But since the order of succession \(x_{j_{1}},\ldots,x_{j_{n}}\) is chronologically correct, the ordinary contractions coincide with the chronological ones. We have arrived at the conclusion that

\[ T\bigl(A_{1}(x_{1})\ldots A_{n}(x_{n})\bigr) \]

is equal to the sum, multiplied by \(\varepsilon\), of the normal products of the operators

\[ A_{j_{1}}(x_{j_{1}})\ldots A_{j_{n}}(x_{j_{n}}) \]

with all possible chronological contractions.

As already noted, under the sign of chronological contraction, as also under the sign of a normal product, one may permute (with due regard for the change of sign) linear operators. Thus, under the sign of normal products with all possible chronological contractions, we may restore the normal order of the factors \(1,2,\ldots,n\), at the same time omitting the factor \(\varepsilon\). The theorem is proved.

Let us introduce into consideration the \(T\)-product of several normal products of field linear operators \(A_{i}(x)\ldots D_{j}(z)\ldots\)

\[ T\bigl(:A_{1}(x)A_{2}(x)\ldots A_{n}(x):\ldots :D_{1}(z)\ldots D_{m}(z):\bigr). \tag{5.11} \]

It is precisely such \(T\)-products that are needed for expanding \(T\)-products of local operators, since by definition

a local operator \(L(x)\) is represented as a linear combination of terms of the type

\[ :A_1(x) A_2(x)\ldots A_n(x): . \]

For \(T\)-products of the form (5.11), the formulation of Wick’s theorem has only the special feature that mutual chronological pairings of operators entering into one and the same normal product must not be taken into account.

Let us note in conclusion that, starting from (5.11), one can also define the \(T\)-product of normal products of a more general form

\[ T\bigl(:A_1(x_1)\ldots A_n(x_n):\ldots :D_1(z_1)\ldots D_m(z_m):\bigr) \tag{5.12} \]

as the sum of normal products of the operators

\[ A_1(x_1)\ldots D_m(z_m) \]

with all possible pairings, excluding mutual pairings of operators standing in one and the same normal product.

On the basis of this definition of the \(T\)-product (5.12), one can introduce the \(T\)-product of polylocal operators

\[ T\bigl(A_1(x_1,\ldots,x_n),\ldots,D(z_1,\ldots,z_m)\bigr), \tag{5.13} \]

representing it as a linear combination of expressions of type (5.12). A direct definition of the \(T\)-product (5.13) according to the chronological criterion is inconvenient in the present case because of the multiplicity of arguments of the operators \(A,\ldots,D\).

It is evident from this that, in essence, the \(T\)-product is a new algebraic operation, which can be introduced independently of ordinary products.

From the mathematical point of view, the \(T\)-product is especially attractive because, unlike ordinary products, under the sign of the \(T\)-product one may permute operators as if they exactly commuted or anticommuted.

§ 6. FEYNMAN RULES FOR CALCULATING MATRIX ELEMENTS AND TRANSITION PROBABILITIES

The Wick theorem presented in the preceding paragraph makes it possible to formulate a convenient prescription for reducing to normal form the \(T\)-products of Lagrangians, which are operator expressions \(S_\nu(x_1,\ldots,x_\nu)\) entering into the scattering matrix. For greater clarity it is convenient to begin the exposition with some concrete case. We shall therefore consider the interaction of electro-

magnetic and spinor electron–positron fields, whose interaction Lagrangian has the form*)

\[ L(x)=\sqrt{4\pi e}\sum_m :\bar{\psi}(x)\gamma^m\psi(x)A_m(x): . \tag{6.1} \]

From Wick’s theorem for \(T\)-products it follows that the coefficient \(S_n(x_1,\ldots,x_n)\) (up to the factor \(i^n\)) is equal to the sum of normal products of \(n\) Lagrangians (6.1)

\[ :L(x_1)\ldots L(x_n): \tag{6.2} \]

with all possible chronological contractions of the operators \(A\), \(\psi\), and \(\bar{\psi}\) entering into \(L(x)\).

Associating with the corresponding contraction of spinor operators

\[ \overline{\psi_\alpha(x_1)\bar{\psi}_\beta(x_2)} =\frac{1}{i}S^c_{\alpha\beta}(x_1-x_2) \tag{6.3} \]

an electron line directed from the point \(x_2\) to the point \(x_1\), and with the electromagnetic contraction

\[ \overline{A_m(x_1)A_n(x_2)} =ig^{mn}D_0^c(x_1-x_2) \tag{6.4} \]

an undirected photon line connecting the points \(x_1\) and \(x_2\), with the matrix \(\gamma^m\) from the Lagrangian \(L(x_l)\) the point \(x_l\) at which two electron lines and one photon line meet, and, finally, with the uncontracted operators \(\bar{\psi}(x_i)\), \(\psi(x_j)\), and \(A(x_k)\), respectively, an external electron line leaving the point \(x_i\), an external electron line entering the point \(x_j\), and an external photon line connected with the point \(x_k\), we obtain the well-known rules of correspondence for constructing the so-called Feynman diagrams corresponding to the elements \(S_n(x_1,\ldots,x_n)\).

These rules must also be supplemented by a sign rule, taking into account the anticommutativity of the spinor operators \(\psi\) and \(\bar{\psi}\). If one agrees to write these operators and their contractions in the order from left to right corresponding to motion opposite to the direction of the electron lines, i.e. in the form

\[ :\bar{\psi}(x_1)\overline{\psi(x_1)\bar{\psi}(x_1)}\ldots \overline{\psi(x_{j_k-1})\bar{\psi}(x_{j_k})}\psi(x_{j_k}): , \tag{6.5} \]

it is not difficult to conclude that expressions of this type must be multiplied by the factor

\[ \varepsilon=(-1)^{\,n-l-m}, \tag{6.5a} \]

*) The factor \(\sqrt{4\pi}\) owes its appearance to the normalization of the electromagnetic-field potential used by us (compare also (6.7), (6.8)).

where \(n\) is the order of the diagram, \(l\) is the number of closed spinor cycles, and \(m\) is the number of open spinor cycles.

Recalling also the factors \(e^n\) and \(i^n(n!)-1\), we arrive at a simple method for constructing the terms of the expansion of the \(S\)-matrix in the \(x\)-representation, which, as is clear, reduces to constructing, according to the rules given above, the operator expressions corresponding to all possible Feynman diagrams of a given order.

Let us note that in this case the form and number of unpaired operators depend on the external lines of the diagrams, while the structure of the coefficient functions is completely determined by the internal parts of these diagrams.

The method described for constructing the terms of the \(S\)-matrix by means of Feynman rules and diagrams can be applied to any other local interaction. In this case the topology of the diagrams is completely determined by the structure of the terms constituting the interaction Lagrangian, while the correspondence rules are determined by the matrix structure of these terms and by the type of pairings of the operators entering into them.

Let us now consider the procedure for calculating the matrix elements of the scattering matrix, which plays an important role in computations of the effective cross sections of various scattering processes and transformations of particles.

For the calculation of matrix elements it is convenient to pass to the momentum representation. This is because, on the one hand, the causal functions (chronological pairings) have a simple structure in the momentum representation, and, on the other hand, matrix elements, as a rule, are taken between states containing particles with fixed momenta.

For clarity of this statement, let us write the momentum expansions of the electron-positron and electromagnetic fields and of the corresponding pairings. For the operators of the electromagnetic field we have:

\[ A_m(x)=A_m^{(+)}(x)+A_m^{(-)}(x), \tag{6.6} \]

\[ A_m^{(\pm)}(x)=\frac{1}{(2\pi)^{3/2}}\int e^{\pm ikx}\delta(k^2)\theta(k^0)A_m^{(\pm)}(k)\,dk, \tag{6.7} \]

\[ A_m^{(\pm)}(k)=\sum_{\nu=0,1,2,3} e_\nu^{\,m} a_\nu^{(\pm)}(k), \tag{6.8} \]

where

\[ \theta(x)= \begin{cases} 1, & x>0,\\ 0, & x<0. \end{cases} \]

Here \(e_r\) are polarization vectors, and \(a_r^{(+)}(k)\) and \(a_r^{(-)}(k)\) are, respectively, the creation and annihilation operators of a photon with polarization \(r\) and 4-momentum \(k\) \((k^2=0)\).

For the electron-positron field

\[ \psi(x)=\psi^{(+)}(x)+\psi^{(-)}(x), \tag{6.9} \]

\[ \bar{\psi}(x)=\bar{\psi}^{(+)}(x)+\bar{\psi}^{(-)}(x), \tag{6.10} \]

\[ \psi^{(\pm)}(x)=\frac{1}{(2\pi)^{7/2}}\int e^{\pm ipx}\delta(p^2-m^2)\theta(p^0)\psi^{(\pm)}(p)\,dp, \tag{6.11} \]

\[ \psi_{\sigma}^{(\pm)}(p)=\sum_{\mu=1,2}u_{\sigma}^{\pm,\mu}(p)a_{\pm}^{\mu}(p), \tag{6.12} \]

\[ \bar{\psi}^{(\pm)}(x)=\frac{1}{(2\pi)^{7/2}}\int e^{\pm ipx}\delta(p^2-m^2)\theta(p^0)\bar{\psi}^{(\pm)}(p)\,dp, \tag{6.13} \]

\[ \bar{\psi}_{\sigma}^{(\pm)}(p)=\sum_{\mu}\bar{u}_{\sigma}^{\pm,\mu}(p)a_{\pm}^{*\mu}(p). \tag{6.14} \]

Here \(\mu\) is the spin variable, \(u^{\pm,\mu}\) and \(\bar{u}^{\pm,\mu}\) are spinor amplitudes, \(a^{+}\) and \(a^{-}\) are the creation and annihilation operators of electrons, and \(a^{+}\) and \(a^{-}\) are the analogous operators for positrons.

Finally, for the contractions,

\[ \overline{A_m(x)A_n(y)}=ig^{mn}D_0^c(x-y) = -\frac{g^{mn}}{(2\pi)^4 i}\int \frac{e^{ik(x-y)}}{k^2+i\varepsilon}\,dk, \tag{6.15} \]

\[ \overline{\psi_{\alpha}(x)\bar{\psi}_{\beta}(y)} = \frac{1}{i}S_{\alpha\beta}^{c}(x-y) = \]

\[ =\frac{i}{(2\pi)^4}\int \frac{\hat{p}+m}{p^2-m^2+i\varepsilon} e^{-ip(x-y)}\,dp, \tag{6.16} \]

\[ \hat{p}=\gamma p=\gamma^0p^0-\boldsymbol{\gamma}\mathbf{p}. \tag{6.17} \]

Owing to the presence of factors of the type \(\delta(k^2-m^2)\theta(k^0)\) in (6.7), (6.11), and (6.13), the integration over \(k^0\) (or \(p^0\)) can be carried out trivially. As a result we obtain an expression of the form

\[ \psi^{(\pm)}(x)=\frac{1}{(2\pi)^{3/2}}\int e^{\pm ipx} \frac{\psi^{(\pm)}(\mathbf{p})}{\sqrt{2p^0}}\,d\mathbf{p} \quad \left(p^0=\sqrt{\mathbf{p}^2+m^2}\right), \tag{6.18} \]

\[ \psi^{(\pm)}(\mathbf{p})=\sum_{\mu}u^{\pm,\mu}(\mathbf{p})a_{\pm}^{\mu}(\mathbf{p}). \tag{6.19} \]

where

\[ u^{\pm,\mu}(\mathbf{p})=u^{\pm,\mu}(p)\big|_{p^0=\sqrt{\mathbf{p}^2+m^2}}, \]

the “three-dimensional” operators \(a(\mathbf p)\) are expressed in terms of the “four-dimensional” operators \(a(p)\) by the relations

\[ a^{\pm}(\mathbf p)= \left. \frac{a^{\pm}(p)}{\sqrt{2p^{0}}} \right|_{p^{0}=+\sqrt{\mathbf p^{2}+m^{2}}} = \int a^{\pm}(p)\sqrt{2p^{0}}\,\delta(p^{2}-m^{2})\,\theta(p^{0})\,dp^{0} \tag{6.20} \]

and obey commutation relations of the type

\[ \bigl[a^{+}(\mathbf p),\overset{*}{a}{}^{-}(\mathbf p')\bigr] = \delta(\mathbf p-\mathbf p'). \tag{6.21} \]

The amplitude of a state containing \(m\) particles of different kinds with definite values of the momenta will now be represented by an expression of the form

\[ \Phi_{\ldots k \ldots} = a_{1}^{(+)}(k_{1})\,a_{2}^{(+)}(k_{2})\ldots a_{m}^{(+)}(k_{m})\Phi_{\mathrm{vac}}. \tag{6.22} \]

In calculating the matrix element

\[ \bigl(\overset{*}{\Phi}_{\ldots k'\ldots},\ldots u_i(x_i)\ldots;\Phi_{\ldots k\ldots}\bigr) \tag{6.23} \]

the creation operators \(u^{(+)}\) must be commuted with the annihilation operators \(a^{(-)}(k')\) from the amplitude \(\overset{*}{\Phi}_{\ldots k'\ldots}\), and the operators \(u^{(-)}\) with the operators \(a^{(+)}(k)\) from the amplitude \(\Phi_{\ldots k\ldots}\), until one of them, acting on \(\overset{*}{\Phi}_{\mathrm{vac}}\) or \(\Phi_{\mathrm{vac}}\), gives zero. This process was considered briefly in § 2.

Restricting ourselves to the case when for none of the particles is the momentum in the initial state equal to the momentum in the final state, we arrive at the conclusion that the matrix element (6.23) is represented as the product of the results of the commutations of the operators

\[ u^{(-)}(x_j)\quad \text{with}\quad a^{(+)}(k) \]

and

\[ u^{(+)}(x_i)\quad \text{with}\quad a^{(-)}(k'), \]

equal to the expression

\[ \prod_{(k)} \frac{u^{(-)}(k)}{(2\pi)^{3/2}}\,e^{-ikx_j} \prod_{(k')} \frac{u^{(+)}(k')}{(2\pi)^{3/2}}\,e^{ik'x_i}. \tag{6.24} \]

The essential fact is that to each external line of the diagram, from the point of view of matrix elements, there corresponds a real particle in the initial or final state. This circumstance also permits Feynman diagrams to be regarded as schematic representations of processes of interaction of elementary particles.

Using the expansion of causal functions of the type (6.15), (6.16) and expression (6.24), we can now carry out, in the terms of the \(S\)-matrix, the integration over the variables \(x\), after which we obtain Feynman rules for calculating the matrix elements of the scattering matrix

\[ \left(\Phi_{\ldots k' \ldots}^{*}\, S(1)\Phi_{\ldots k \ldots}\right). \tag{6.25} \]

For example, for spinor electrodynamics, in the notation used by us, we obtain the following correspondence rules (table):

Table

Particle	Diagram element	Factor in the matrix element
1. Electron in the initial state	line with momentum \(p\)	\((2\pi)^{-3/2} u^{-,\nu}(\mathbf p)\)
2. Positron in the initial state	line with momentum \(p\)	\((2\pi)^{-3/2} \bar u^{-,\nu}(\mathbf p)\)
3. Electron in the final state	line with momentum \(p\)	\((2\pi)^{-3/2} \bar u^{+,\nu}(\mathbf p)\)
4. Positron in the final state	line with momentum \(p\)	\((2\pi)^{-3/2} u^{+,\nu}(\mathbf p)\)
5. Photon in the initial or final states with polarization \(e_n\)	wavy line with momentum \(k\)	\(\displaystyle \frac{e_n^{\,m}}{(2\pi)^{3/2}\sqrt{2k^0}},\quad n=1,2\)
6. Virtual electron with momentum \(p\), moving from \(1\) to \(2\), or virtual positron with momentum \(p\), moving from \(2\) to \(1\)	fermion line with momentum \(p\) from \(1\) to \(2\)	\(\displaystyle \frac{i}{(2\pi)^4}\,\frac{\hat p+m}{p^2-m^2+i\varepsilon}\)
7. Virtual photon with momentum \(k\)	wavy line with momentum \(k\)	\(\displaystyle \frac{g^{mn}}{(2\pi)^4 i}\,\frac{1}{k^2+i\varepsilon}\)
8. Diagram vertex	vertex with \(p_1\), \(p_2\), \(k\)	\(\displaystyle \sqrt{4\pi\alpha}\,\gamma^m(2\pi)^4\delta(p_2-p_1\pm k)\)

When constructing matrix elements in the momentum representation it is also necessary to take into account the symmetry properties of the functions \(S_n(x_1,\ldots,x_n)\). This property leads to the fact that when among the \(n\) vertices of a diagram there are \(k\) groups of \(v_1,v_2,\ldots,v_k\) vertices,

entering symmetrically, the entire expression must be multiplied by the factor

\[ \frac{n!}{\nu_1!\nu_2!\ldots \nu_k!}. \]

One must also note the case when, in the initial and final states, there are several particles of one and the same kind (several photons, several electrons, etc.). In this case the creation and annihilation operators from the amplitudes \(\dot{\Phi}\) and \(\Phi\) may be commuted with the operators from the scattering matrix in several different ways. The corresponding diagrams will differ by a permutation of the momenta of external lines of identical particles. In doing this, one must also take into account that the normalization amplitude of a state containing \(k\) groups of \(N_1,\ldots,N_k\) identical particles contains the factor

\[ (N_1!N_2!\ldots N_k!)^{-1/2} \]

and that, in the case of identical fermions, the resulting matrix element must be antisymmetric with respect to the permutation of any pair of Fermi particles.

As was established, the matrix elements (6.25) are functions of the initial and final momenta \(p\) and \(p'\) and, by virtue of the correspondence rules, are proportional to the \(\delta\)-function

\[ \delta\left(\sum p-\sum p'\right), \]

expressing the general law of conservation of 4-momentum, i.e.,

\[ (\dot{\Phi}_{\ldots p'\ldots}S(1)\Phi_{\ldots p\ldots}) = \delta\left(\sum p-\sum p'\right)F(p',p). \tag{6.26} \]

In determining the probabilities of scattering processes one has to compute squares of matrix elements of the type

\[ \left|(\dot{\Phi}_{\ldots p'\ldots}S(1)\Phi_{\ldots p\ldots})\right|^2, \]

whose direct evaluation with the aid of (6.26) leads to a meaningless expression. A more detailed consideration*) shows that in this case we obtain

\[ \left|(\dot{\Phi}_{\ldots p'\ldots}S(1)\Phi_{\ldots p\ldots})\right|^2 = \frac{VT}{(2\pi)^4} \delta\left(\sum p-\sum p'\right) |F(p',p)|^2, \tag{6.27} \]

where \(V\) and \(T\) are the spatial volume and the time interval in which the interaction occurs.

*) See, for example, 10.

In the particular case of scattering of particles by a classical stationary field, when momentum is not conserved but only energy is conserved, we have:

\[ (\dot{\Phi}_{\ldots p'\ldots} S(1)\Phi_{\ldots p\ldots})= \delta\left(\sum E-\sum E'\right)F(p',p), \tag{6.28} \]

\[ |(\dot{\Phi}_{\ldots p'\ldots} S(1)\Phi_{\ldots p\ldots})|^2= \frac{T}{2\pi}\,\delta\left(\sum E-\sum E'\right)|F(p',p)|^2 . \tag{6.29} \]

Let us establish, finally, the connection between the matrix elements of the \(S\)-matrix and transition probabilities.

Considering the amplitude of a one-particle state with fixed momentum \(p_0\) as the limit of the amplitude of a wave packet of particles with momenta lying in a small region about \(p_0\) as this region is narrowed, one can show that the amplitude of a one-particle state normalized to unit volume has the form

\[ \Phi=(2\pi)^{3/2}a^+(p_0)\Phi_{\mathrm{vac}}. \]

Therefore the normalized amplitude of the initial state, containing \(s\) particles with definite momenta, may be written in the form

\[ \Phi=(2\pi)^{\frac{3s}{2}}a_1^+(p_1)\ldots a_s^+(p_s)\Phi_{\mathrm{vac}} =(2\pi)^{\frac{3s}{2}}\Phi_{\ldots p\ldots}. \tag{6.30} \]

Choosing the amplitude of the final state \(\Phi_a\) in the form

\[ \Phi_a=\int_G \Phi_{\ldots k\ldots}\,dk_1\ldots dk_s\,(2\pi)^{\frac{3s}{2}}, \tag{6.31} \]

integrated over a region \(G\) equal to the product of the volumes

\[ \Delta p'_1\ldots \Delta p'_r, \]

we obtain the probability of transition from the state (6.30) to the state (6.31), according to the general rules of quantum mechanics, in the form

\[ dw=\frac{(\dot{\Phi}_a S(1)\Phi_0)^2}{(\dot{\Phi}_a\Phi_a)}= \]

\[ =(2\pi)^{3s}\Delta p'_1\ldots \Delta p'_r |(\dot{\Phi}_{\ldots p'\ldots}S(1)\Phi_{\ldots p\ldots})|^2. \tag{6.32} \]

Here it is assumed, in connection with the choice of normalization, that in the initial state the mean numbers of particles per unit volume are equal to unity. If, however, these numbers are respectively equal to \(n_1,\ldots,n_s\), then expression (6.32) should be replaced by

\[ dw=n_1\ldots n_s(2\pi)^{3s}\Delta p'_1\ldots \Delta p'_r |(\dot{\Phi}_{\ldots p'\ldots}(S)(1)\Phi_{\ldots p\ldots})|^2. \tag{6.33} \]

As an illustration, let us consider the process of scattering of a particle by a particle. In this case \(r=s=2\). Instead of (6.33), taking (6.27) into account, we obtain the transition probability per unit time and per unit volume

\[ (2\pi)^2 n_1 n_2 \delta(p_1+p_2-p'_1-p'_2)|F(p',p)|^2\,dp'_1\,dp'_2 . \]

Since \(p'_2\) is completely determined by \(p'_1\) and the initial momenta, carrying out the integration over \(p'_2\), we find:

\[ (2\pi)^2 n_1 n_2 |F(p',p)|^2 \delta(p^0_1+p^0_2-p'{}^0_1-p'{}^0_2)\,dp'_1, \tag{6.34} \]

where in \(F\) one has put

\[ p'_2=p_1+p_2-p'_1 . \]

Expression (6.34) is usually represented in the form of the product

\[ n_1 n_2 v_1\,d\sigma_1, \]

where \(v_1\) is the modulus of the velocity of the particle with momentum \(p_1\), equal to

\[ v_1=\frac{|\mathbf p_1|}{p^0_1}, \]

and \(d\sigma_1\) has the dimension of area, is proportional to the element of solid angle of the momentum of particle \(p'_1\) after scattering, is called the differential effective cross section, and has the form

\[ d\sigma_1= \frac{(2\pi)^2}{v_1} |F(p',p)|^2 \delta(E_1+E_2-E'_1-E'_2)\,dp'_1\,p_1'^2\,d\Omega'_1, \tag{6.35} \]

where it is denoted

\[ E_i=p_i^0,\qquad p'_i=|\mathbf p'_i|. \]

Passing from \(p'_1\) to \(E'_1\) by means of the relation

\[ E_2'^2=p_1'^2+m^2 \]

and carrying out the integration over \(E'_1\), we find the final formula

\[ d\sigma_1= \frac{(2\pi)^2 p'_1 E'_1}{v_1|f'(E'_1)|} |F(p',p)|^2 d\Omega'_1\Big|_{p_1+p_2-p'_1-p'_2=0}, \tag{6.36} \]

where it is denoted

\[ f'(E'_1)=\frac{d}{dE'_1}(E'_1+E'_2-E_2-E_1). \]

The total effective scattering cross section is obtained by integrating (6.36) over $\Omega'$:

\[ \sigma_1=\frac{(2\pi)^3}{v_1}\int_{\Omega'}\frac{p'_1 E'_1}{\left|f'(E'_1)\right|}\,\left|F(p',p)\right|^2\,d\Omega'. \tag{6.37} \]

Let us also note that usually, when calculating cross sections, one is not interested in the spin states of the particles. Therefore, summation is carried out over the spin indices of the scattered particles, and averaging over the spin indices of the scattering particles. Denoting these operations briefly by the symbol $\sum_\sigma$, we obtain the formulas for the differential and total effective cross section in the form

\[ d\sigma_1=\frac{(2\pi)^3 p'_1 E'_1}{v_1\left|f'(E'_1)\right|}\sum_\sigma \left|F(p',p)\right|^2 d\Omega'_1 \Big|_{p_1+p_2-p'_1-p'_2=0}, \tag{6.38} \]

\[ \sigma_1=\frac{(2\pi)^3}{v_1}\int_{\Omega'}\frac{p'_1 E'_1}{\left|f'(E'_1)\right|}\sum_\sigma \left|F(p',p)\right|^2 d\Omega' \Big|_{p_1+p_2-p'_1-p'_2=0}. \tag{6.39} \]

In a similar way, from the general formula (6.33) one can obtain expressions for the probabilities and cross sections of other possible processes.

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